Integration by Parts

The Integration-by-Parts Formula If, h(x)=f(x)g(x), then by using the product rule, we obtain h′(x)=f′(x)g(x)+g′(x)f(x). Although at first it may seem counterproductive, let’s now integrate both sides of this equation: ∫h′(x)dx=∫(g(x)f′(x)+f(x)g′(x))dx. This gives us h(x)=f(x)g(x)=∫g(x)f′(x)dx+∫f(x)g′(x)dx. Now we solve for ∫f(x)g′(x)dx: ∫f(x)g′(x)dx=f(x)g(x)−∫g(x)f′(x)dx. By making the substitutions u=f(x) and v=g(x), which in turn make du=f′(x)dx and dv=g′(x)dx, we have the more compact form ∫udv=uv−∫vdu. Integration by Parts Let u=f(x) and v=g(x) be functions with continuous derivatives. Then, the integration-by-parts formula for the integral involving these two functions is: ∫udv=uv−∫vdu. The advantage of using the integration-by-parts formula is that we can use it to exchange one integral for another, possibly easier, integral. The following example illustrates its use. Using Integration by Parts Use integration by parts with u=x and dv=sinxdx to evaluate ∫xsinxdx. By choosing u=x, we have du=1dx. Since dv=sinxdx, we get v=∫sinxdx=−cosx. It is handy to keep track of these values as follows: u=xdv=sinxdxdu=1dxv=∫sinxdx=−cosx. Applying the integration-by-parts formula results in ∫xsinxdx=(x)(−cosx)−∫(−cosx)(1dx)Substitute.=−xcosx+∫cosxdxSimplify.=−xcosx+sinx+C.Use∫cosxdx=sinx+C. Analysis At this point, there are probably a few items that need clarification. First of all, you may be curious about what would have happened if we had chosen u=sinx and dv=x. If we had done so, then we would have du=cosxdx and v=12x2. Thus, after applying integration by parts, we have ∫xsinxdx=12x2sinx−∫12x2cosxdx. Unfortunately, with the new integral, we are in no better position than before. It is important to keep in mind that when we apply integration by parts, we may need to try several choices for u and dv before finding a choice that works. Second, you may wonder why, when we find v=∫sinxdx=−cosx, we do not use v=−cosx+K. To see that it makes no difference, we can rework the problem using v=−cosx+K: ∫xsinxdx=(x)(−cosx+K)−∫(−cosx+K)(1dx)=−xcosx+Kx+∫cosxdx−∫Kdx=−xcosx+Kx+sinx−Kx+C=−xcosx+sinx+C. As you can see, it makes no difference in the final solution. Last, we can check to make sure that our antiderivative is correct by differentiating −xcosx+sinx+C: ddx(−xcosx+sinx+C)=(−1)cosx+(−x)(−sinx)+cosx=xsinx. Therefore, the antiderivative checks out. Watch this video and visit this website for examples of integration by parts. Evaluate ∫xe2xdx using the integration-by-parts formula with u=x and dv=e2xdx. ∫xe2xdx=12xe2x−14e2x+C Hint Find du and v, and use the previous example as a guide. The natural question to ask at this point is: How do we know how to choose u and dv? Sometimes it is a matter of trial and error; however, the acronym LIATE can often help to take some of the guesswork out of our choices. This acronym stands for Logarithmic Functions, Inverse Trigonometric Functions, Algebraic Functions, Trigonometric Functions, and Exponential Functions. This mnemonic serves as an aid in determining an appropriate choice for u. The type of function in the integral that appears first in the list should be our first choice of u. For example, if an integral contains a logarithmic function and an algebraic function, we should choose u to be the logarithmic function, because L comes before A in LIATE. The integral in has a trigonometric function (sinx) and an algebraic function (x). Because A comes before T in LIATE, we chose u to be the algebraic function. When we have chosen u, dv is selected to be the remaining part of the function to be integrated, together with dx. Why does this mnemonic work? Remember that whatever we pick to be dv must be something we can integrate. Since we do not have integration formulas that allow us to integrate simple logarithmic functions and inverse trigonometric functions, it makes sense that they should not be chosen as values for dv. Consequently, they should be at the head of the list as choices for u. Thus, we put LI at the beginning of the mnemonic. (We could just as easily have started with IL, since these two types of functions won’t appear together in an integration-by-parts problem.) The exponential and trigonometric functions are at the end of our list because they are fairly easy to integrate and make good choices for dv. Thus, we have TE at the end of our mnemonic. (We could just as easily have used ET at the end, since when these types of functions appear together it usually doesn’t really matter which one is u and which one is dv.) Algebraic functions are generally easy both to integrate and to differentiate, and they come in the middle of the mnemonic. Using Integration by Parts Evaluate ∫lnxx3dx. Begin by rewriting the integral: ∫lnxx3dx=∫x−3lnxdx. Since this integral contains the algebraic function x−3 and the logarithmic function lnx, choose u=lnx, since L comes before A in LIATE. After we have chosen u=lnx, we must choose dv=x−3dx. Next, since u=lnx, we have du=1xdx. Also, v=∫x−3dx=−12x−2. Summarizing, u=lnxdv=x−3dxdu=1xdxv=∫x−3dx=−12x−2. Substituting into the integration-by-parts formula () gives ∫lnxx3dx=∫x−3lnxdx=(lnx)(−12x−2)−∫(−12x−2)(1xdx)=−12x−2lnx+∫12x−3dxSimplify.=−12x−2lnx−14x−2+CIntegrate.=−12x2lnx−14x2+C.Rewrite with positive integers. Evaluate ∫xlnxdx. 12x2lnx−14x2+C Hint Use u=lnx and dv=xdx. In some cases, as in the next two examples, it may be necessary to apply integration by parts more than once. Applying Integration by Parts More Than Once Evaluate ∫x2e3xdx. Using LIATE, choose u=x2 and dv=e3xdx. Thus, du=2xdx and v=∫e3xdx=(13)e3x. Therefore, u=x2dv=e3xdxdu=2xdxv=∫e3xdx=13e3x. Substituting into produces ∫x2e3xdx=13x2e3x−∫23xe3xdx. We still cannot integrate ∫23xe3xdx directly, but the integral now has a lower power on x. We can evaluate this new integral by using integration by parts again. To do this, choose u=x and dv=23e3xdx. Thus, du=dx and v=∫(23)e3xdx=(29)e3x. Now we have u=xdv=23e3xdxdu=dxv=∫23e3xdx=29e3x. Substituting back into the previous equation yields ∫x2e3xdx=13x2e3x−(29xe3x−∫29e3xdx). After evaluating the last integral and simplifying, we obtain ∫x2e3xdx=13x2e3x−29xe3x+227e3x+C. Applying Integration by Parts When LIATE Doesn’t Quite Work Evaluate ∫t3et2dt. If we use a strict interpretation of the mnemonic LIATE to make our choice of u, we end up with u=t3 and dv=et2dt. Unfortunately, this choice won’t work because we are unable to evaluate ∫et2dt. However, since we can evaluate ∫tet2dt, we can try choosing u=t2 and dv=tet2dt. With these choices we have u=t2dv=tet2dtdu=2tdtv=∫tet2dt=12et2. Thus, we obtain ∫t3et2dt=12t2et2−∫12et22tdt=12t2et2−12et2+C. Applying Integration by Parts More Than Once Evaluate ∫sin(lnx)dx. This integral appears to have only one function—namely, sin(lnx) —however, we can always use the constant function 1 as the other function. In this example, let’s choose u=sin(lnx) and dv=1dx. (The decision to use u=sin(lnx) is easy. We can’t choose dv=sin(lnx)dx because if we could integrate it, we wouldn’t be using integration by parts in the first place!) Consequently, du=(1/x)cos(lnx)dx and v=∫1dx=x. After applying integration by parts to the integral and simplifying, we have ∫sin(lnx)dx=xsin(lnx)−∫cos(lnx)dx. Unfortunately, this process leaves us with a new integral that is very similar to the original. However, let’s see what happens when we apply integration by parts again. This time let’s choose u=cos(lnx) and dv=1dx, making du=−(1/x)sin(lnx)dx and v=∫1dx=x. Substituting, we have ∫sin(lnx)dx=xsin(lnx)−(xcos(lnx)—∫−sin(lnx)dx). After simplifying, we obtain ∫sin(lnx)dx=xsin(lnx)−xcos(lnx)−∫sin(lnx)dx. The last integral is now the same as the original. It may seem that we have simply gone in a circle, but now we can actually evaluate the integral. To see how to do this more clearly, substitute I=∫sin(lnx)dx. Thus, the equation becomes I=xsin(lnx)−xcos(lnx)−I. First, add I to both sides of the equation to obtain 2I=xsin(lnx)−xcos(lnx). Next, divide by 2: I=12xsin(lnx)−12xcos(lnx). Substituting I=∫sin(lnx)dx again, we have ∫sin(lnx)dx=12xsin(lnx)−12xcos(lnx). From this we see that (1/2)xsin(lnx)−(1/2)xcos(lnx) is an antiderivative of sin(lnx)dx. For the most general antiderivative, add +C: ∫sin(lnx)dx=12xsin(lnx)−12xcos(lnx)+C. Analysis If this method feels a little strange at first, we can check the answer by differentiation: ddx(12xsin(lnx)−12xcos(lnx))=12(sin(lnx))+cos(lnx)·1x·12x−(12cos(lnx)−sin(lnx)·1x·12x)=sin(lnx). Evaluate ∫x2sinxdx. −x2cosx+2xsinx+2cosx+C Hint This is similar to .

Integration by Parts for Definite Integrals Now that we have used integration by parts successfully to evaluate indefinite integrals, we turn our attention to definite integrals. The integration technique is really the same, only we add a step to evaluate the integral at the upper and lower limits of integration. Integration by Parts for Definite Integrals Let u=f(x) and v=g(x) be functions with continuous derivatives on [a,b]. Then ∫abudv=uv|ab−∫abvdu. Finding the Area of a Region Find the area of the region bounded above by the graph of y=tan−1x and below by the x-axis over the interval [0,1]. This region is shown in . To find the area, we must evaluate ∫01tan−1xdx.

For this integral, let’s choose u=tan−1x and dv=dx, thereby making du=1x2+1dx and v=x. After applying the integration-by-parts formula () we obtain Area=xtan−1x|01−∫01xx2+1dx. Use u-substitution to obtain ∫01xx2+1dx=12ln|x2+1|01. Thus, Area=xtan−1x|01−12ln|x2+1||01=π4−12ln2. At this point it might not be a bad idea to do a “reality check” on the reasonableness of our solution. Since π4−12ln2≈0.4388, and from we expect our area to be slightly less than 0.5, this solution appears to be reasonable. Finding a Volume of Revolution Find the volume of the solid obtained by revolving the region bounded by the graph of f(x)=e−x, the x-axis, the y-axis, and the line x=1 about the y-axis. The best option to solving this problem is to use the shell method. Begin by sketching the region to be revolved, along with a typical rectangle (see the following graph).

To find the volume using shells, we must evaluate 2π∫01xe−xdx. To do this, let u=x and dv=e−x. These choices lead to du=dx and v=∫e−x=−e−x. Substituting into , we obtain Volume=2π∫01xe−xdx=2π(−xe−x|01+∫01e−xdx)Use integration by parts.=−2πxe−x|01−2πe−x|01Evaluate∫01e−xdx=−e−x|01.=2π−4πe.Evaluate and simplify. Analysis Again, it is a good idea to check the reasonableness of our solution. We observe that the solid has a volume slightly less than that of a cylinder of radius 1 and height of 1/e added to the volume of a cone of base radius 1 and height of 1−1e. Consequently, the solid should have a volume a bit less than π(1)21e+(π3)(1)2(1−1e)=2π3e+π3≈1.8177. Since 2π−4πe≈1.6603, we see that our calculated volume is reasonable. Evaluate ∫0π/2xcosxdx. π2−1 Hint Use with u=x and dv=cosxdx.

In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals. ∫x3e2xdx u=x3 ∫x3ln(x)dx ∫y3cosydy u=y3 ∫x2arctanxdx ∫e3xsin(2x)dx u=sin(2x) Find the integral by using the simplest method. Not all problems require integration by parts. ∫vsinvdv ∫lnxdx (Hint: ∫lnxdx is equivalent to ∫1·ln(x)dx.) −x+xlnx+C ∫xcosxdx ∫tan−1xdx xtan−1x−12ln(1+x2)+C ∫x2exdx ∫xsin(2x)dx −12xcos(2x)+14sin(2x)+C ∫xe4xdx ∫xe−xdx e−x(−1−x)+C ∫xcos3xdx ∫x2cosxdx 2xcosx+(−2+x2)sinx+C ∫xlnxdx ∫ln(2x+1)dx 12(1+2x)(−1+ln(1+2x))+C ∫x2e4xdx ∫exsinxdx 12ex(−cosx+sinx)+C ∫excosxdx ∫xe−x2dx −e−x22+C ∫x2e−xdx ∫sin(ln(2x))dx −12xcos[ln(2x)]+12xsin[ln(2x)]+C ∫cos(lnx)dx ∫(lnx)2dx 2x−2xlnx+x(lnx)2+C ∫ln(x2)dx ∫x2lnxdx (−x39+13x3lnx)+C ∫sin−1xdx ∫cos−1(2x)dx −121−4x2+xcos−1(2x)+C ∫xarctanxdx ∫x2sinxdx −(−2+x2)cosx+2xsinx+C ∫x3cosxdx ∫x3sinxdx −x(−6+x2)cosx+3(−2+x2)sinx+C ∫x3exdx ∫xsec−1xdx 12x(−1−1x2+x·sec−1x)+C ∫xsec2xdx ∫xcoshxdx −coshx+xsinhx+C Compute the definite integrals. Use a graphing utility to confirm your answers. ∫1/e1lnxdx ∫01xe−2xdx (Express the answer in exact form.) 14−34e2 ∫01exdx(letu=x) ∫1eln(x2)dx 2 ∫0πxcosxdx ∫−ππxsinxdx (Express the answer in exact form.) 2π ∫03ln(x2+1)dx (Express the answer in exact form.) ∫0π/2x2sinxdx (Express the answer in exact form.) −2+π ∫01x5xdx (Express the answer using five significant digits.) Evaluate ∫cosxln(sinx)dx −sin(x)+ln[sin(x)]sinx+C Derive the following formulas using the technique of integration by parts. Assume that n is a positive integer. These formulas are called reduction formulas because the exponent in the x term has been reduced by one in each case. The second integral is simpler than the original integral. ∫xnexdx=xnex−n∫xn−1exdx ∫xncosxdx=xnsinx−n∫xn−1sinxdx Answers vary ∫xnsinxdx=______ Integrate ∫2x2x−3dx using two methods: Using parts, letting dv=2x−3dx Substitution, letting u=2x−3 a. 25(1+x)(−3+2x)3/2+C b. 25(1+x)(−3+2x)3/2+C State whether you would use integration by parts to evaluate the integral. If so, identify u and dv. If not, describe the technique used to perform the integration without actually doing the problem. ∫xlnxdx ∫ln2xxdx Do not use integration by parts. Choose u to be lnx, and the integral is of the form ∫u2du. ∫xexdx ∫xex2−3dx Do not use integration by parts. Let u=x2−3, and the integral can be put into the form ∫eudu. ∫x2sinxdx ∫x2sin(3x3+2)dx Do not use integration by parts. Choose u to be u=3x3+2 and the integral can be put into the form ∫sin(u)du. Sketch the region bounded above by the curve, the x-axis, and x=1, and find the area of the region. Provide the exact form or round answers to the number of places indicated. y=2xe−x (Approximate answer to four decimal places.) y=e−xsin(πx) (Approximate answer to five decimal places.) The area under graph is 0.39535.

Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated. y=sinx,y=0,x=2π,x=3π about the y-axis (Express the answer in exact form.) y=e−x y=0,x=−1x=0; about x=1 (Express the answer in exact form.) 2πe A particle moving along a straight line has a velocity of v(t)=t2e−t after t sec. How far does it travel in the first 2 sec? (Assume the units are in feet and express the answer in exact form.) Find the area under the graph of y=sec3x from x=0tox=1. (Round the answer to two significant digits.) 2.05 Find the area between y=(x−2)ex and the x-axis from x=2 to x=5. (Express the answer in exact form.) Find the area of the region enclosed by the curve y=xcosx and the x-axis for 11π2≤x≤13π2. (Express the answer in exact form.) 12π Find the volume of the solid generated by revolving the region bounded by the curve y=lnx, the x-axis, and the vertical line x=e2 about the x-axis. (Express the answer in exact form.) Find the volume of the solid generated by revolving the region bounded by the curve y=4cosx and the x-axis, π2≤x≤3π2, about the x-axis. (Express the answer in exact form.) 8π2 Find the volume of the solid generated by revolving the region in the first quadrant bounded by y=ex and the x-axis, from x=0 to x=ln(7), about the y-axis. (Express the answer in exact form.)