# Calculus for AP

## Educators

MK

Problem 1

evaluate the integral using the Integration by Parts formula with the given choice of $u$ and $v^{\prime} .$ $$\int x \sin x d x ; \quad u=x, v^{\prime}=\sin x$$

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Problem 2

evaluate the integral using the Integration by Parts formula with the given choice of $u$ and $v^{\prime} .$ $$\int x e^{2 x} d x ; \quad u=x, v^{\prime}=e^{2 x}$$

MK
Margaret K.

Problem 3

evaluate the integral using the Integration by Parts formula with the given choice of $u$ and $v^{\prime} .$ $$\int(2 x+9) e^{x} d x ; \quad u=2 x+9, v^{\prime}=e^{x}$$

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Problem 4

evaluate the integral using the Integration by Parts formula with the given choice of $u$ and $v^{\prime} .$$\int x \cos 4 x d x ; \quad u=x, v^{\prime}=\cos 4 x$$ MK Margaret K. Numerade Educator Problem 5 evaluate the integral using the Integration by Parts formula with the given choice of$u$and$v^{\prime} .$$\int x^{3} \ln x d x ; \quad u=\ln x, v^{\prime}=x^{3}$$

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Problem 6

evaluate the integral using the Integration by Parts formula with the given choice of $u$ and $v^{\prime} .$$\int \tan ^{-1} x d x ; \quad u=\tan ^{-1} x, v^{\prime}=1$$ MK Margaret K. Numerade Educator Problem 7 evaluate using Integration by Parts. $$\int(4 x-3) e^{-x} d x$$ Check back soon! Problem 8 evaluate using Integration by Parts.$$\int(2 x+1) e^{x} d x$$ MK Margaret K. Numerade Educator Problem 9 evaluate using Integration by Parts.$$\int x e^{5 x+2} d x$$ Check back soon! Problem 10 evaluate using Integration by Parts$$\int x^{2} e^{x} d x$$ MK Margaret K. Numerade Educator Problem 11 evaluate using Integration by Parts$$\int x \cos 2 x d x$$ Check back soon! Problem 12 evaluate using Integration by Parts$$\int x \sin (3-x) d x$$ MK Margaret K. Numerade Educator Problem 13 evaluate using Integration by Parts$$\int x^{2} \sin x d x$$ Check back soon! Problem 14 evaluate using Integration by Parts$$\int x^{2} \cos 3 x d x$$ MK Margaret K. Numerade Educator Problem 15 evaluate using Integration by Parts$$\int e^{-x} \sin x d x$$ Check back soon! Problem 16 evaluate using Integration by Parts$$\int e^{x} \sin 2 x d x$$ MK Margaret K. Numerade Educator Problem 17 evaluate using Integration by Parts$$\int e^{-5 x} \sin x d x$$ Check back soon! Problem 18 evaluate using Integration by Parts$$\int e^{3 x} \cos 4 x d x$$ MK Margaret K. Numerade Educator Problem 19 evaluate using Integration by Parts$$\int x \ln x d x$$ Check back soon! Problem 20 evaluate using Integration by Parts$$\int \frac{\ln x}{x^{2}} d x$$ MK Margaret K. Numerade Educator Problem 21 evaluate using Integration by Parts$$\int x^{2} \ln x d x$$ Check back soon! Problem 22 evaluate using Integration by Parts$$\int x^{-5} \ln x d x$$ MK Margaret K. Numerade Educator Problem 23 evaluate using Integration by Parts$$\int(\ln x)^{2} d x$$ Check back soon! Problem 24 evaluate using Integration by Parts$$\int x(\ln x)^{2} d x$$ MK Margaret K. Numerade Educator Problem 25 evaluate using Integration by Parts$$\int x \sec ^{2} x d x$$ Check back soon! Problem 26 evaluate using Integration by Parts$$\int x \tan x \sec x d x$$ MK Margaret K. Numerade Educator Problem 27 evaluate using Integration by Parts$$\int \cos ^{-1} x d x$$ Check back soon! Problem 28 evaluate using Integration by Parts$$\int \sin ^{-1} x d x$$ MK Margaret K. Numerade Educator Problem 29 evaluate using Integration by Parts$$\int \sec ^{-1} x d x$$ Check back soon! Problem 30 evaluate using Integration by Parts$$\int x 5^{x} d x$$ MK Margaret K. Numerade Educator Problem 31 evaluate using Integration by Parts$$\int 3^{x} \cos x d x$$ Check back soon! Problem 32 evaluate using Integration by Parts$$\int x \sinh x d x$$ MK Margaret K. Numerade Educator Problem 33 evaluate using Integration by Parts$$\int x^{2} \cosh x d x$$ Check back soon! Problem 34 evaluate using Integration by Parts$$\int \cos x \cosh x d x$$ MK Margaret K. Numerade Educator Problem 35 evaluate using Integration by Parts$$\int \tanh ^{-1} 4 x d x$$ Check back soon! Problem 36 evaluate using Integration by Parts$$\int \sinh ^{-1} x d x$$ MK Margaret K. Numerade Educator Problem 37 evaluate using substitution and then Integration by Parts. $$\int e^{\sqrt{x}} d x$$ Hint: Let $$u=x^{1 / 2} \quad 38 . \int x^{3} e^{x^{2}} d x$$ Check back soon! Problem 38 evaluate using substitution and then Integration by Parts.$$\int x^{3} e^{x^{2}} d x$$ MK Margaret K. Numerade Educator Problem 39 evaluate using Integration by Parts, substitution, or both if necessary. $$\int x \cos 4 x d x$$ Check back soon! Problem 40 evaluate using Integration by Parts, substitution, or both if necessary. $$\int \frac{\ln (\ln x) d x}{x}$$ MK Margaret K. Numerade Educator Problem 41 evaluate using Integration by Parts, substitution, or both if necessary. $$\int \frac{x d x}{\sqrt{x+1}}$$ Check back soon! Problem 42 evaluate using Integration by Parts, substitution, or both if necessary. $$\int x^{2}\left(x^{3}+9\right)^{15} d x$$ MK Margaret K. Numerade Educator Problem 43 evaluate using Integration by Parts, substitution, or both if necessary.$$\int \cos x \ln (\sin x) d x$$ Check back soon! Problem 44 evaluate using Integration by Parts, substitution, or both if necessary.$$\int \sin \sqrt{x} d x$$ MK Margaret K. Numerade Educator Problem 45 evaluate using Integration by Parts, substitution, or both if necessary$$\int \sqrt{x} e^{\sqrt{x}} d x$$ Check back soon! Problem 46 evaluate using Integration by Parts, substitution, or both if necessary$$\int \frac{\tan \sqrt{x} d x}{\sqrt{x}}$$ MK Margaret K. Numerade Educator Problem 47 evaluate using Integration by Parts, substitution, or both if necessary$$\int \frac{\ln (\ln x) \ln x d x}{x}$$ Check back soon! Problem 48 evaluate using Integration by Parts, substitution, or both if necessary$$\int \sin (\ln x) d x$$ Check back soon! Problem 49 compute the definite integral. $$\int_{0}^{3} x e^{4 x} d x$$ Check back soon! Problem 50 compute the definite integral. $$\int_{0}^{\pi / 4} x \sin 2 x d x$$ MK Margaret K. Numerade Educator Problem 51 compute the definite integral. $$\int_{1}^{2} x \ln x d x$$ Check back soon! Problem 52 compute the definite integral. $$\int_{1}^{e} \frac{\ln x d x}{x^{2}}$$ MK Margaret K. Numerade Educator Problem 53 compute the definite integral. $$\int_{0}^{\pi} e^{x} \sin x d x$$ Check back soon! Problem 54 compute the definite integral. $$\int_{0}^{1} \tan ^{-1} x d x$$ Check back soon! Problem 55 Use Eq. (5) to evaluate$\int x^{4} e^{x} d x$Check back soon! Problem 56 Use substitution and then Eq. (5) to evaluate$\int x^{4} e^{7 x} d x$Check back soon! Problem 57 Find a reduction formula for$\int x^{n} e^{-x} d x$similar to Eq.$(5)$Check back soon! Problem 58 Evaluate$\int x^{n} \ln x d x$for$n \neq-1 .$Which method should be used to evaluate$\int x^{-1} \ln x d x ?$Check back soon! Problem 59 In Exercises$59-66$, indicate a good method for evaluating the integral but do not evaluate). Your choices are algebraic manpulation, substitution (specify$u$and$d u ),$and Integration by Parts (specify u and$v^{\prime}$). If it appears that the techniques you have learned thus fare not suffcient, state this. $$\int \sqrt{x} \ln x d x$$ Check back soon! Problem 60 In Exercises$59-66$, indicate a good method for evaluating the integral but do not evaluate). Your choices are algebraic manpulation, substitution (specify$u$and$d u ),$and Integration by Parts (specify u and$v^{\prime}$). If it appears that the techniques you have learned thus fare not suffcient, state this. $$\int \frac{x^{2}-\sqrt{x}}{2 x} d x$$ Check back soon! Problem 61 In Exercises$59-66$, indicate a good method for evaluating the integral but do not evaluate). Your choices are algebraic manpulation, substitution (specify$u$and$d u ),$and Integration by Parts (specify u and$v^{\prime}$). If it appears that the techniques you have learned thus fare not suffcient, state this. $$\int \frac{x^{3} d x}{\sqrt{4-x^{2}}}$$ Check back soon! Problem 62 In Exercises$59-66$, indicate a good method for evaluating the integral but do not evaluate). Your choices are algebraic manpulation, substitution (specify$u$and$d u ),$and Integration by Parts (specify u and$v^{\prime}$). If it appears that the techniques you have learned thus fare not suffcient, state this. $$\int \frac{d x}{\sqrt{4-x^{2}}}$$ Check back soon! Problem 63 In Exercises$59-66$, indicate a good method for evaluating the integral but do not evaluate). Your choices are algebraic manpulation, substitution (specify$u$and$d u ),$and Integration by Parts (specify u and$v^{\prime}$). If it appears that the techniques you have learned thus fare not suffcient, state this. $$\int \frac{x+2}{x^{2}+4 x+3} d x$$ Check back soon! Problem 64 In Exercises$59-66$, indicate a good method for evaluating the integral but do not evaluate). Your choices are algebraic manpulation, substitution (specify$u$and$d u ),$and Integration by Parts (specify u and$v^{\prime}$). If it appears that the techniques you have learned thus fare not suffcient, state this. $$\int \frac{d x}{(x+2)\left(x^{2}+4 x+3\right)}$$ Check back soon! Problem 65 In Exercises$59-66$, indicate a good method for evaluating the integral but do not evaluate). Your choices are algebraic manpulation, substitution (specify$u$and$d u ),$and Integration by Parts (specify u and$v^{\prime}$). If it appears that the techniques you have learned thus fare not suffcient, state this. $$\int x \sin (3 x+4) d x$$ Check back soon! Problem 66 In Exercises$59-66$, indicate a good method for evaluating the integral but do not evaluate). Your choices are algebraic manpulation, substitution (specify$u$and$d u ),$and Integration by Parts (specify u and$v^{\prime}$). If it appears that the techniques you have learned thus fare not suffcient, state this. $$\int x \cos \left(9 x^{2}\right) d x$$ Check back soon! Problem 67 Evaluate$\int\left(\sin ^{-1} x\right)^{2} d x .$Hint: Use Integration by Parts first and then substitution. Check back soon! Problem 68 Evaluate$\int \frac{(\ln x)^{2} d x}{x^{2}} .$Hint: Use substitution first and then Integration by Parts. Check back soon! Problem 69 Evaluate $$\int x^{7} \cos \left(x^{4}\right) d x$$ Check back soon! Problem 70 Find$f(x),$assuming that $$\int f(x) e^{x} d x=f(x) e^{x}-\int x^{-1} e^{x} d x$$ Check back soon! Problem 71 Find the volume of the solid obtained by revolving the region under$y=e^{x}$for$0 \leq x \leq 2$about the$y$-axis. Check back soon! Problem 72 Find the area enclosed by$y=\ln x$and$y=(\ln x)^{2}$Check back soon! Problem 73 Recall that the present value (PV) of an investment that pays out income continuously at a rate$R(t)$for$T$years is$\int_{0}^{T} R(t) e^{-r t} d t$where$r$is the interest rate. Find the PV if$R(t)=5000+100 t$S/year,$r=0.05$and$T=10$years. Check back soon! Problem 74 Derive the reduction formula $$\int(\ln x)^{k} d x=x(\ln x)^{k}-k \int(\ln x)^{k-1} d x$$ Check back soon! Problem 75 Use Eq. (6) to calculate$\int(\ln x)^{k} d x$for$k=2,3$Check back soon! Problem 76 Derive the reduction formulas $$\int x^{n} \cos x d x=x^{n} \sin x-n \int x^{n-1} \sin x d x$$ $$\int x^{n} \sin x d x=-x^{n} \cos x+n \int x^{n-1} \cos x d x$$ Check back soon! Problem 77 Prove that $$\int x b^{x} d x=b^{x}\left(\frac{x}{\ln b}-\frac{1}{\ln ^{2} b}\right)+C$$ Check back soon! Problem 78 Define$P_{n}(x)$by $$\int x^{n} e^{x} d x=P_{n}(x) e^{x}+C$$ Use Eq.$(5)$to prove that$P_{n}(x)=x^{n}-n P_{n-1}(x) .$Use this recursion relation to find$P_{n}(x)$for$n=1,2,3,4$. Note that$P_{0}(x)=1$Check back soon! Problem 79 The Integration by Parts formula can be written $$\int u(x) v(x) d x=u(x) V(x)-\int u^{\prime}(x) V(x) d x$$ where$V(x)$satisfies$V^{\prime}(x)=v(x)$$$\begin{array}{l}{\text { (a) Show directly that the right-hand side of Eq. (7) does not change }} \\ {\text { if } V(x) \text { is replaced by } V(x)+C, \text { where } C \text { is a constant. }}\end{array}$$ $$\begin{array}{l}{\text { (b) Use } u=\tan ^{-1} x \text { and } v=x \text { in Eq. }(7) \text { to calculate } \int x \tan ^{-1} x d x} \\ {\text { but carry out the calculation twice: first with } V(x)=\frac{1}{2} x^{2} \text { and then with }} \\ {V(x)=\frac{1}{2} x^{2}+\frac{1}{2} . \text { Which choice of } V(x) \text { results in a simpler calculation? }}\end{array}$$ Check back soon! Problem 80 Prove in two ways that $$\int_{0}^{a} f(x) d x=a f(a)-\int_{0}^{a} x f^{\prime}(x) d x$$ First use Integration by Parts. Then assume$f(x)$is increasing. Use the substitution$u=f(x)$to prove that$\int_{0}^{a} x f^{\prime}(x) d x$is equal to the area of the shaded region in Figure 1 and derive Eq.$(8)$a second time. Check back soon! Problem 81 Assume that$f(0)=f(1)=0$and that$f^{\prime \prime}$exists. Prove $$\int_{0}^{1} f^{\prime \prime}(x) f(x) d x=-\int_{0}^{1} f^{\prime}(x)^{2} d x$$ Use this to prove that if$f(0)=f(1)=0$and$f^{\prime \prime}(x)=\lambda f(x)$for some constant$\lambda,$then$\lambda<0 .$Can you think of a function satisfying these conditions for some$\lambda$? Check back soon! Problem 82 Set$I(a, b)=\int_{0 } ^ {1} x ^ {a}(1-x) ^ {b} d x,$where$a, b$are whole numbers. $$\begin{array}{c}{\text { (a) Use substitution to show that } I(a, b)=I(b, a) \text { . }} \\ {\text { (b) Show that } I(a, 0)=I(0, a)=\frac{1}{a+1}} \\ {\text { (c) Prove that for } a \geq 1 \text { and } b \geq 0} \\ {I(a, b)=\frac{a}{b+1} I(a-1, b+1)}\end{array}$$ $$\begin{array}{l}{\text { (d) Use (b) and (c) to calculate } I(1,1) \text { and } I(3,2) \text { . }} \\ {\text { (e) Show that } I(a, b)=\frac{a ! b !}{(a+b+1) !}}\end{array}$$ Check back soon! Problem 83 Let$I_{n}=\int x^{n} \cos \left(x^{2}\right) d x$and$J_{n}=\int x^{n} \sin \left(x^{2}\right) d x$$$\begin{array}{l}{\text { (a) Find a reduction formula that expresses } I_{n} \text { in terms of } J_{n-2} . \text { Hint: }} \\ {\text { Write } x^{n} \cos \left(x^{2}\right) \text { as } x^{n-1}\left(x \cos \left(x^{2}\right)\right)}\end{array}$$ $$\begin{array}{l}{\text { (b) } \text { Use the result of (a) to show that } I_{n} \text { can be evaluated explicitly if } n \text { is odd. }}\end{array}$$ (c) Evaluate$I_{3}\$

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