How to Integrate

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11-11-2020, 05:10
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It is commonly said that differentiation is a science while integration is an art. The reason is because integration is simply a harder task to do—while a derivative is only concerned with the behavior of a function at a point, an integral, being a glorified sum, integration requires global knowledge of the function. So while there are some functions whose integrals can be evaluated using the standard techniques in this article, many more cannot.

The Basics

  1. Use integration to find areas, central points, and volumes. You can use integration to find many useful parts of a graph that you’ve made. However, it’s usually the easiest to start by finding the area under the curve of a function on a graph.
    • Integration is most often used in conjunction with the quadratic equation.
  2. Memorize the notation for integration. An integral ∫abf(x)dx{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x} consists of four parts:
    • The ∫{\displaystyle \int } is the symbol for integration. It is actually an elongated S.
    • The function f(x){\displaystyle f(x)} is called the integrand when it is inside the integral.
    • The differential dx{\displaystyle \mathrm {d} x} intuitively is saying what variable you are integrating with respect to. Because (Riemann) integration is just a sum of infinitesimally thin rectangles with a height of f(x),{\displaystyle f(x),} we see that dx{\displaystyle \mathrm {d} x} refers to the width of those rectangles.
    • The letters a{\displaystyle a} and b{\displaystyle b} are the boundaries. An integral does not need to have boundaries. When this is the case, we say that we are dealing with an indefinite integral. If it does, then we are dealing with a definite integral.
    • Throughout this article, we will go over the process of finding antiderivatives of a function. An antiderivative is a function whose derivative is the original function we started with.
  3. Understand the definition of an integral. When we talk about integrals, we usually refer to Riemann integrals; in other words, summing up rectangles. Given a function f(x),{\displaystyle f(x),} a rectangle width of Δx,{\displaystyle \Delta x,} and an interval [a,b],{\displaystyle [a,b],} the area of the first rectangle is given by f(x1)Δx1,{\displaystyle f(x_{1})\Delta x_{1},} because it is just the base times the height (the value of the function). Similarly, the area of the second rectangle is f(x2)Δx2.{\displaystyle f(x_{2})\Delta x_{2}.} Generalizing, we say the area of the ith rectangle is f(xi)Δxi.{\displaystyle f(x_{i})\Delta x_{i}.} You can represent this by saying:
    • ∑i=1nf(xi)Δxi{\displaystyle \sum _{i=1}^{n}f(x_{i})\Delta x_{i}}
    • If this is the first time you have seen a summation symbol, it may look scary—but it's not complicated at all. All this is saying is that we are summing up the area of n{\displaystyle n} rectangles. (The variable i{\displaystyle i} is known as a dummy index.)
    • However, as you can guess, the area of all the rectangles is bound to be slightly different from the true area. We solve this by sending the number of rectangles to infinity. As we increase the number of rectangles, the area of all the rectangles better approximates the area under the curve. The limit as n→∞{\displaystyle n\to \infty } is what we define as the integral of the function f(x){\displaystyle f(x)} from a{\displaystyle a} to b.{\displaystyle b.}
      • ∫abf(x)dx=limn→∞∑i=1nf(xi)Δxi{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x=\lim _{n\to \infty }\sum _{i=1}^{n}f(x_{i})\Delta x_{i}}
    • Of course, this limit has to exist in order for the integral to have any meaning. If such a limit does not exist on the interval, then we say that f(x){\displaystyle f(x)} does not have an integral over the interval [a,b].{\displaystyle [a,b].} In this article (and in almost every physical application), we only deal with functions where these integrals exist.
  4. Remember +C{\displaystyle +C} when evaluating indefinite integrals. One of the most common mistakes people can make is forgetting to add the constant of integration. The reason why this is needed is because antiderivatives are not unique. In fact, a function can have an infinite number of antiderivatives. They are allowed because the derivative of a constant is 0.

Power Rule

  1. The Power Rule: This rule lets you find both the indefinite and definite integrals. Just take the exponent in the function and add 1, then divide by that same value. Finally, add the constant C.{\displaystyle C.}
    • You can use this rule in many functions, like polynomials, functions with roots, and rational functions. Simply put, if you can write it with exponents, you can most likely use the power rule.
  2. Consider a monomial xn{\displaystyle x^{n}}, then perform the power rule. This is the same power rule for derivatives, but in reverse. We increase the power by 1, and divide by the new power. Don't forget to add the constant of integration C.{\displaystyle C.}
    • ∫xndx=xn+1n+1+C{\displaystyle \int x^{n}\mathrm {d} x={\frac {x^{n+1}}{n+1}}+C}
    • To verify that this power rule holds, differentiate the antiderivative to recover the original function.
    • The power rule holds for all functions of this form with degree n{\displaystyle n} except when n=−1.{\displaystyle n=-1.} We will see why later.
  3. Apply linearity. Integration is a linear operator, which means that the integral of a sum is the sum of the integrals, and the coefficient of each term can be factored out, like so:
    • ∫(axn+bxm)dx=a∫xndx+b∫xmdx{\displaystyle \int (ax^{n}+bx^{m})\mathrm {d} x=a\int x^{n}\mathrm {d} x+b\int x^{m}\mathrm {d} x}
    • This should be familiar because the derivative is also a linear operator; the derivative of a sum is the sum of the derivatives.
    • Linearity does not apply just for integrals of polynomials. It applies to any integral where the integrand is a sum of two or more terms.
  4. Find the antiderivative of the function f(x)=x4+2x3−5x2−1{\displaystyle f(x)=x^{4}+2x^{3}-5x^{2}-1}. This is a polynomial, so using the property of linearity and the power rule, the antiderivative can easily be computed. To find the antiderivative of a constant, remember that x0=1,{\displaystyle x^{0}=1,} so the constant is really just the coefficient of x0.{\displaystyle x^{0}.}
    • ∫(x4+2x3−5x2−1)dx=15x5+24x4−53x3−x+C{\displaystyle \int (x^{4}+2x^{3}-5x^{2}-1)\mathrm {d} x={\frac {1}{5}}x^{5}+{\frac {2}{4}}x^{4}-{\frac {5}{3}}x^{3}-x+C}
  5. Find the antiderivative of the function f(x)=2x2+3x−1x1/3{\displaystyle f(x)={\frac {2x^{2}+3x-1}{x^{1/3}}}}. This may seem like a function that defies our rules, but a moment's glance reveals that we can separate the fraction into three fractions and apply linearity and the power rule to find the antiderivative.
    • ∫f(x)dx=∫2x2+3x−1x1/3dx=∫(2x2x1/3+3xx1/3−1x1/3)dx=∫(2x5/3+3x2/3−x−1/3)dx=2⋅38x8/3+3⋅35x5/3−32x2/3+C=34x8/3+95x5/3−32x2/3+C{\displaystyle {\begin{aligned}\int f(x)\mathrm {d} x&=\int {\frac {2x^{2}+3x-1}{x^{1/3}}}\mathrm {d} x\\&=\int \left({\frac {2x^{2}}{x^{1/3}}}+{\frac {3x}{x^{1/3}}}-{\frac {1}{x^{1/3}}}\right)\mathrm {d} x\\&=\int (2x^{5/3}+3x^{2/3}-x^{-1/3})\mathrm {d} x\\&=2\cdot {\frac {3}{8}}x^{8/3}+3\cdot {\frac {3}{5}}x^{5/3}-{\frac {3}{2}}x^{2/3}+C\\&={\frac {3}{4}}x^{8/3}+{\frac {9}{5}}x^{5/3}-{\frac {3}{2}}x^{2/3}+C\end{aligned}}}
    • The common theme is that you must perform whatever manipulations in order to get the integral into a polynomial. From there, integration is easy. Judging whether the integral is easy enough to brute-force, or requires some algebraic manipulation first, is where the skill lies.

Definite Integration

  1. Definite Integral: If an integral is definite, it has both start and end values. Therefore, it would be written like this: (a,b){\displaystyle (a,b)}. When you write them in an equation, you put them on the top and the bottom of the long S, like this:.
    • ∫ab{\displaystyle \int _{a}^{b}}
    • If the long S does not have any numbers on the top or bottom, it’s an indefinite integral.
  2. Consider the integral below. Notice how there are numbers on both the top and the bottom of the long S. This means that unlike the integration process in part 2, we also have bounds to evaluate at.
    • ∫23x2dx{\displaystyle \int _{2}^{3}x^{2}\mathrm {d} x}
  3. Use the fundamental theorem of calculus. This theorem is in two parts. The first part was stated in the first sentence of this article: integration is the inverse operation of differentiation, so integrating and then differentiating a function recovers the original function. The second part is stated below.
    • Let F(x){\displaystyle F(x)} be an antiderivative of f(x).{\displaystyle f(x).} Then ∫abf(x)dx=F(b)−F(a).{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x=F(b)-F(a).}
    • This theorem is incredibly useful because it simplifies the integral and means that the definite integral is completely determined by just the values on its boundaries. There is no need to sum up rectangles anymore to compute integrals. All we need to do now is to find antiderivatives, and evaluate at the bounds!
  4. Evaluate the original integral. Now that we have the fundamental theorem as a tool for solving integrals, we can easily calculate the value of the integral as defined above.
    • ∫23x2dx=13x3|23=13(3)3−13(2)3=193{\displaystyle {\begin{aligned}\int _{2}^{3}x^{2}\mathrm {d} x&={\frac {1}{3}}x^{3}{\Bigg |}_{2}^{3}\\&={\frac {1}{3}}(3)^{3}-{\frac {1}{3}}(2)^{3}\\&={\frac {19}{3}}\end{aligned}}}
    • Again, the fundamental theorem of calculus does not just apply to functions like f(x)=x2.{\displaystyle f(x)=x^{2}.} The fundamental theorem can be used to integrate any function, as long as you can find an antiderivative.
  5. Remember to keep the boundaries in place when you evaluate. If you evaluate the integral with the boundaries swapped, you’ll get the inverse answer. Let's try it out to see what happens:
    • ∫32x2dx=13x3|32=13(2)3−13(3)3=−193{\displaystyle {\begin{aligned}\int _{3}^{2}x^{2}\mathrm {d} x&={\frac {1}{3}}x^{3}{\Bigg |}_{3}^{2}\\&={\frac {1}{3}}(2)^{3}-{\frac {1}{3}}(3)^{3}\\&=-{\frac {19}{3}}\end{aligned}}}
    • We just obtained the negative of the answer we got before. This illustrates an important property of definite integrals. Swapping the boundaries negates the integral.
      • ∫baf(x)dx=−∫abf(x)dx{\displaystyle \int _{b}^{a}f(x)\mathrm {d} x=-\int _{a}^{b}f(x)\mathrm {d} x}

Antiderivatives of Common Functions

  1. Antiderivative: Finding the antiderivative of a function means finding the function that has a derivative equal to our original function. Often, antiderivatives get mixed up with integration, but they are slightly different.
    • The definition sounds paradoxical, but it will become clearer with an example.
  2. Memorize the antiderivatives of exponential functions. In the following steps, we list commonly encountered functions like the exponential and trigonometric functions. All are widely encountered, so knowing what their antiderivatives are is crucial to building up integrating skills. Remember that indefinite integrals have an extra C,{\displaystyle C,} because the derivative of a constant is 0.
    • ∫exdx=ex+C{\displaystyle \int e^{x}\mathrm {d} x=e^{x}+C}
    • ∫axdx=axln⁡a+C{\displaystyle \int a^{x}\mathrm {d} x={\frac {a^{x}}{\ln a}}+C}
  3. Memorize the antiderivatives of trigonometric functions. These are just the derivatives applied backwards and should be familiar. The sines and cosines are encountered far more often and should definitely be memorized. Hyperbolic analogues are similarly found, though they are encountered less often.
    • ∫sin⁡xdx=−cos⁡x+C{\displaystyle \int \sin x\mathrm {d} x=-\cos x+C}
    • ∫cos⁡xdx=sin⁡x+C{\displaystyle \int \cos x\mathrm {d} x=\sin x+C}
    • ∫sec2⁡xdx=tan⁡x+C{\displaystyle \int \sec ^{2}x\mathrm {d} x=\tan x+C}
    • ∫csc2⁡xdx=−cot⁡x+C{\displaystyle \int \csc ^{2}x\mathrm {d} x=-\cot x+C}
    • ∫sec⁡xtan⁡xdx=sec⁡x+C{\displaystyle \int \sec x\tan x\mathrm {d} x=\sec x+C}
    • ∫csc⁡xcot⁡xdx=−csc⁡x+C{\displaystyle \int \csc x\cot x\mathrm {d} x=-\csc x+C}
  4. Memorize the antiderivatives of inverse trigonometric functions. These should not really be considered an exercise in "memorization." As long as you are familiar with the derivatives, then most of these antiderivatives should be familiar as well.
    • ∫11−x2dx=sin−1⁡x+C{\displaystyle \int {\frac {1}{\sqrt {1-x^{2}}}}\mathrm {d} x=\sin ^{-1}x+C}
    • ∫−11−x2dx=cos−1⁡x+C{\displaystyle \int {\frac {-1}{\sqrt {1-x^{2}}}}\mathrm {d} x=\cos ^{-1}x+C}
    • ∫11+x2dx=tan−1⁡x+C{\displaystyle \int {\frac {1}{1+x^{2}}}\mathrm {d} x=\tan ^{-1}x+C}
  5. Memorize the antiderivative of the reciprocal function. Previously, we said that the function f(x)=x−1,{\displaystyle f(x)=x^{-1},} or f(x)=1x,{\displaystyle f(x)={\frac {1}{x}},} was an exception to the power rule. The reason is because the antiderivative of this function is the logarithmic function.
    • ∫1xdx=ln⁡|x|+C{\displaystyle \int {\frac {1}{x}}\mathrm {d} x=\ln |x|+C}
    • (Sometimes, authors like to put the dx{\displaystyle \mathrm {d} x} in the numerator of the fraction, so it reads like ∫dxx.{\displaystyle \int {\frac {\mathrm {d} x}{x}}.} Be aware of this notation.)
    • The reason for the absolute value in the logarithm function is subtle, and requires a more thorough understanding of real analysis in order to fully answer. For now, we will just live with the fact that the domains become the same when the absolute value bars are added.
  6. Evaluate the following integral over the given bounds. Our function is given as f(x)=2cos⁡x+tan2⁡x−6.{\displaystyle f(x)=2\cos x+\tan ^{2}x-6.} Here, we don't know the antiderivative of tan2⁡x,{\displaystyle \tan ^{2}x,} but we can use a trigonometric identity to rewrite the integrand in terms of a function which we know the antiderivative of - namely, 1+tan2⁡x=sec2⁡x.{\displaystyle 1+\tan ^{2}x=\sec ^{2}x.}
    • ∫π/4π/3f(x)dx=∫π/4π/3(2cos⁡x+tan2⁡x−6)dx=∫π/4π/3(2cos⁡x+sec2⁡x−7)dx=(2sin⁡x+tan⁡x−7x)|π/4π/3=(2sin⁡π3+tan⁡π3−7π3)−(2sin⁡π4+tan⁡π4−7π4)=23−2−1−7π12{\displaystyle {\begin{aligned}\int _{\pi /4}^{\pi /3}f(x)\mathrm {d} x&=\int _{\pi /4}^{\pi /3}(2\cos x+\tan ^{2}x-6)\mathrm {d} x\\&=\int _{\pi /4}^{\pi /3}(2\cos x+\sec ^{2}x-7)\mathrm {d} x\\&=(2\sin x+\tan x-7x){\Big |}_{\pi /4}^{\pi /3}\\&=\left(2\sin {\frac {\pi }{3}}+\tan {\frac {\pi }{3}}-{\frac {7\pi }{3}}\right)-\left(2\sin {\frac {\pi }{4}}+\tan {\frac {\pi }{4}}-{\frac {7\pi }{4}}\right)\\&=2{\sqrt {3}}-{\sqrt {2}}-1-{\frac {7\pi }{12}}\end{aligned}}}
    • If you need a decimal approximation, you can use a calculator here. 23−2−1−7π12≈−0.7827.{\displaystyle 2{\sqrt {3}}-{\sqrt {2}}-1-{\frac {7\pi }{12}}\approx -0.7827.}

Integrals of Symmetric Functions

  1. Evaluate the integral of an even function. Even functions are functions with the property that f(−x)=f(x).{\displaystyle f(-x)=f(x).} In other words, you should be able to replace every x{\displaystyle x} with a −x{\displaystyle -x} and get the same function. An example of an even function is x2.{\displaystyle x^{2}.} Another example is the cosine function. All even functions are symmetric about the y-axis.
    • ∫−11(cos⁡x+x4)dx{\displaystyle \int _{-1}^{1}(\cos x+x^{4})\mathrm {d} x}
    • Our integrand is even. We can immediately integrate by using the fundamental theorem of calculus, but if we look more carefully, we see that the bounds are symmetric about x=0.{\displaystyle x=0.} That means that the integral from -1 to 0 is going to give us the same value as the integral from 0 to 1. So what we can do is we can change the bounds to 0 and 1 and factor out a 2.
      • ∫−11(cos⁡x+x4)dx=2∫01(cos⁡x+x4)dx{\displaystyle \int _{-1}^{1}(\cos x+x^{4})\mathrm {d} x=2\int _{0}^{1}(\cos x+x^{4})\mathrm {d} x}
    • It might not seem like much to do this, but we will immediately see that our work is simplified. After finding the antiderivative, notice that we only need to evaluate it at x=1.{\displaystyle x=1.} The antiderivative at x=0{\displaystyle x=0} will not contribute to the integral.
      • 2∫01(cos⁡x+x4)dx=2sin⁡1+25{\displaystyle 2\int _{0}^{1}(\cos x+x^{4})\mathrm {d} x=2\sin 1+{\frac {2}{5}}}
  2. Perform a simplification whenever you see an even function. In general, whenever you see an even function with symmetric boundaries, you should perform this simplification in order to make less arithmetic mistakes. It will make your workload easier so you don’t have to spend as much time correcting yourself at the end.
      • ∫−aaf(x)dx=2∫0af(x)dx,  f(x) even{\displaystyle \int _{-a}^{a}f(x)\mathrm {d} x=2\int _{0}^{a}f(x)\mathrm {d} x,\ \ f(x)\ \mathrm {even} }
  3. Evaluate the integral of an odd function. Odd functions are functions with the property that f(−x)=−f(x).{\displaystyle f(-x)=-f(x).} In other words, you should be able to replace every x{\displaystyle x} with a −x{\displaystyle -x} and then get the negative of the original function. An example of an odd function is x3.{\displaystyle x^{3}.} The sine and tangent functions are also odd. All odd functions are symmetric about the origin (imagine rotating the negative part of the function by 180° - it will then stack on top of the positive part of the function). If the bounds are symmetric, then the integral will be 0.
    • ∫−π/2π/2(2x3+2sin⁡x)dx{\displaystyle \int _{-\pi /2}^{\pi /2}(2x^{3}+2\sin x)\mathrm {d} x}
    • We could evaluate this integral directly...or we can recognize that our integrand is odd. Furthermore, the boundaries are symmetric about the origin. Therefore, our integral is 0. Why is this the case? It is because the antiderivative is even. Even functions have the property that f(−x)=f(x),{\displaystyle f(-x)=f(x),} so when we evaluate at the bounds −a{\displaystyle -a} and a,{\displaystyle a,} then F(−a)=F(a){\displaystyle F(-a)=F(a)} immediately implies that F(a)−F(−a)=0.{\displaystyle F(a)-F(-a)=0.}
      • ∫−π/2π/2(2x3+2sin⁡x)dx=0{\displaystyle \int _{-\pi /2}^{\pi /2}(2x^{3}+2\sin x)\mathrm {d} x=0}
    • The properties of these functions are very potent in simplifying the integrals, but the boundaries must be symmetric. Otherwise, we will need to evaluate the old way.
      • ∫−aaf(x)dx=0,  f(x) odd{\displaystyle \int _{-a}^{a}f(x)\mathrm {d} x=0,\ \ f(x)\ \mathrm {odd} }

U-Substitution

  1. U-substitution: This is a technique that changes variables with the hope of obtaining an easier integral. As we will see, it is the analogue of the chain rule for derivatives.
    • This is also called “Integration by Substitution” or “The Reverse Chain Rule.”
  2. Evaluate the integral of eax{\displaystyle e^{ax}}. What do we do when the exponent has a coefficient in it? We use u-substitution to change variables. It turns out that these kinds of u-subs are the easiest to perform, and they are done so often, the u-sub is often skipped. Nevertheless, we will show the entire process.
    • ∫eaxdx{\displaystyle \int e^{ax}\mathrm {d} x}
  3. Choose a u{\displaystyle u} and find du{\displaystyle \mathrm {d} u}. We choose u=ax{\displaystyle u=ax} so that we get a eu{\displaystyle e^{u}} in the integrand, a function whose antiderivative we are familiar with - itself. Then we must replace dx{\displaystyle \mathrm {d} x} with du,{\displaystyle \mathrm {d} u,} but we need to make sure that we are keeping track of our terms. In this example, du=adx,{\displaystyle \mathrm {d} u=a\mathrm {d} x,} so we need to divide the whole integral by a{\displaystyle a} to compensate.
    • ∫eaxdx=1a∫eudu{\displaystyle \int e^{ax}\mathrm {d} x={\frac {1}{a}}\int e^{u}\mathrm {d} u}
  4. Evaluate and rewrite in terms of the original variable. For indefinite integrals, you must rewrite in terms of the original variable.
    • ∫eaxdx=1aeax+C{\displaystyle \int e^{ax}\mathrm {d} x={\frac {1}{a}}e^{ax}+C}
  5. Evaluate the following integral with the given boundaries. This is a definite integral, so we need to evaluate the antiderivative at the boundaries. We will also see that this u-sub is a case where you need to "back-substitute."
    • ∫01x2x+3dx{\displaystyle \int _{0}^{1}x{\sqrt {2x+3}}\mathrm {d} x}
  6. Choose a u{\displaystyle u} and find du{\displaystyle \mathrm {d} u}. Make sure to change your boundaries as well according to your substitution. We choose u=2x+3{\displaystyle u=2x+3} so that we simplify the square root. Then du=2dx,{\displaystyle \mathrm {d} u=2\mathrm {d} x,} and the bounds then go from 3 to 5. However, after replacing the dx{\displaystyle \mathrm {d} x} with a du,{\displaystyle \mathrm {d} u,} we still have an x{\displaystyle x} in the integrand.
    • ∫01x2x+3dx=12∫35xudu{\displaystyle \int _{0}^{1}x{\sqrt {2x+3}}\mathrm {d} x={\frac {1}{2}}\int _{3}^{5}x{\sqrt {u}}\mathrm {d} u}
  7. Solve for x{\displaystyle x} in terms of u{\displaystyle u} and substitute. This is the back-substitution that we were talking about earlier. Our u-sub did not get rid of all the x{\displaystyle x} terms in the integrand, so we need to back-sub to get rid of it. We find that x=u−32.{\displaystyle x={\frac {u-3}{2}}.} After simplifying, we get the following.
    • 12∫35xudu=14∫35(u−3)udu{\displaystyle {\frac {1}{2}}\int _{3}^{5}x{\sqrt {u}}\mathrm {d} u={\frac {1}{4}}\int _{3}^{5}(u-3){\sqrt {u}}\mathrm {d} u}
  8. Expand and evaluate. An advantage when dealing with definite integrals is that you do not need to rewrite the antiderivative in terms of the original variable before evaluating. Doing so would introduce needless complications.
    • 14∫35(u−3)udu=14∫35(u3/2−3u1/2)du=14(25u5/2−3⋅23u3/2)|35=14(25(5)5/2−2⋅(5)3/2−25(3)5/2+2⋅(3)3/2)=14(−65⋅33/2+2⋅33/2)=144533/2=335{\displaystyle {\begin{aligned}{\frac {1}{4}}\int _{3}^{5}(u-3){\sqrt {u}}\mathrm {d} u&={\frac {1}{4}}\int _{3}^{5}(u^{3/2}-3u^{1/2})\mathrm {d} u\\&={\frac {1}{4}}\left({\frac {2}{5}}u^{5/2}-3\cdot {\frac {2}{3}}u^{3/2}\right){\Bigg |}_{3}^{5}\\&={\frac {1}{4}}\left({\frac {2}{5}}(5)^{5/2}-2\cdot (5)^{3/2}-{\frac {2}{5}}(3)^{5/2}+2\cdot (3)^{3/2}\right)\\&={\frac {1}{4}}\left(-{\frac {6}{5}}\cdot 3^{3/2}+2\cdot 3^{3/2}\right)\\&={\frac {1}{4}}{\frac {4}{5}}3^{3/2}\\&={\frac {3{\sqrt {3}}}{5}}\end{aligned}}}

Integration by Parts

  1. Integrate by Parts: The integration by parts formula is given below. The main goal of integration by parts is to integrate the product of two functions - hence, it is the analogue of the product rule for derivatives. This technique simplifies the integral into one that is hopefully easier to evaluate.
    • ∫udv=uv−∫vdu{\displaystyle \int u\mathrm {d} v=uv-\int v\mathrm {d} u}
  2. Evaluate the integral of the logarithm function. We know that the derivative of ln⁡x{\displaystyle \ln x} is 1x,{\displaystyle {\frac {1}{x}},} but not the antiderivative. It turns out that this integral is a simple application of integration by parts.
    • ∫ln⁡xdx{\displaystyle \int \ln x\mathrm {d} x}
  3. Choose a u{\displaystyle u} and dv,{\displaystyle \mathrm {d} v,} and find du{\displaystyle \mathrm {d} u} and v{\displaystyle v}. We choose u=ln⁡x{\displaystyle u=\ln x} because the derivative is algebraic and therefore easier to manipulate. Then dv=dx.{\displaystyle \mathrm {d} v=\mathrm {d} x.} Therefore, du=1xdx{\displaystyle \mathrm {d} u={\frac {1}{x}}\mathrm {d} x} and v=x.{\displaystyle v=x.} Substituting all of these into the formula, we obtain the following.
    • ∫ln⁡xdx=xln⁡x−∫dx{\displaystyle \int \ln x\mathrm {d} x=x\ln x-\int \mathrm {d} x}
    • We converted the integral of a logarithm into the integral of 1, which is trivial to evaluate.
  4. Evaluate to get your answer. Remember to include C{\displaystyle C} as an addition.
    • ∫ln⁡xdx=xln⁡x−x+C{\displaystyle \int \ln x\mathrm {d} x=x\ln x-x+C}

Using a Calculator

  1. Select “Math,” then “9.” From your home screen, this will take you to the definite integral function. You can use your arrow buttons to scroll up and down on the screen and select which field you’d like to choose.
    • This is a method that works for both TI-83 calculators and TI-84 calculators. Older models may be slightly different.
    • You can only work with definite integrals on the calculator. If the integral doesn’t have any boundaries, you won’t be able to enter it.
    • If you’d like to graph your integral instead of just getting an answer, head to your Y plots instead.
  2. Type in the boundaries and the function. The boundaries are the numbers on the top and the bottom of the long S symbol. Type in the boundaries and the function exactly how you see them in the equation.
    • Remember not to flip the boundaries, as that will give you an inverse answer.
    • If you’re graphing your integrals, type the boundaries and the function into the Y1 plot. Then, select the upper and lower boundaries of the integral as best you can on the graph itself.
  3. Press “Enter” to get your answer. Your calculator will tell you the definite integral in a whole number as your answer. Or, if you’re doing a graph, it will show you what your integral looks like on the screen along with the answer.
    • Remember, this is only for definite integrals.

Tips

  • When defining the Riemann sum, the meaning of f(xi){\displaystyle f(x_{i})} can mean the left-hand, right-hand, or midpoint of f(xi){\displaystyle f(x_{i})} on the interval [xi,xi+Δx].{\displaystyle [x_{i},x_{i}+\Delta x].}
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