Stewart 8th Section 7.1: Integration by Parts

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Stewart 8th Section 7.1: Integration by Parts 0/80
1 Exercise - Integration by Parts (Guided) · Level 1
Evaluate the integral using integration by parts with the indicated choices of \(u\) and \(d v\): \(\int x e^{2x} d x\); \(u = x\), \(d v = e^{2x} d x\).
2 Exercise - Integration by Parts (Guided) · Level 1
Evaluate the integral using integration by parts with the indicated choices of \(u\) and \(d v\): \(\int \sqrt{x} \ln x d x\); \(u = \ln x\), \(d v = \sqrt{x} d x\).
3 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int x \cos 5x d x\).
4 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int y e^{0.2 y} d y\).
5 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int t e^{-3t} d t\).
6 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int (x-1) \sin \pi x d x\).
7 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int (x^2 + 2x) \cos x d x\).
8 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int t^2 \sin \beta t d t\).
9 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int \cos^{-1} x d x\).
10 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int \ln \sqrt{x} d x\).
11 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int t^4 \ln t d t\).
12 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int \tan^{-1} 2y d y\).
13 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int t \csc^2 t d t\).
14 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int x \cosh a x d x\).
15 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int (\ln x)^2 d x\).
16 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int \dfrac{z}{10^z} d z\).
17 Exercise - Integration by Parts · Level 3
Evaluate the integral: \(\int e^{2 \theta} \sin 3 \theta d \theta\).
18 Exercise - Integration by Parts · Level 3
Evaluate the integral: \(\int e^{-\theta} \cos 2 \theta d \theta\).
19 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int z^3 e^z d z\).
20 Exercise - Integration by Parts · Level 2
Evaluate the integral: \(\int x \tan^2 x d x\).
21 Exercise - Integration by Parts · Level 3
Evaluate the integral: \(\int \dfrac{x e^{2x}}{(1+2x)^2} d x\).
22 Exercise - Integration by Parts · Level 3
Evaluate the integral: \(\int (\arcsin x)^2 d x\).
23 Exercise - Definite Integration by Parts · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{1}{2}} x \cos \pi x d x\).
24 Exercise - Definite Integration by Parts · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{1} (x^2 + 1) e^{-x} d x\).
25 Exercise - Definite Integration by Parts · Level 2
Evaluate the integral: \(\displaystyle\int_{0}^{2} y \sinh y d y\).
26 Exercise - Definite Integration by Parts · Level 2
Evaluate the integral: \(\displaystyle\int_{1}^{2} w^2 \ln w d w\).
27 Exercise - Definite Integration by Parts · Level 2
Evaluate the integral: \(\displaystyle\int_{1}^{5} \dfrac{\ln R}{R^2} d R\).
28 Exercise - Definite Integration by Parts · Level 3
Evaluate the integral: \(\displaystyle\int_{0}^{2 \pi} t^2 \sin 2t d t\).
29 Exercise - Definite Integration by Parts · Level 3
Evaluate the integral: \(\displaystyle\int_{0}^{\pi} x \sin x \cos x d x\).
30 Exercise - Definite Integration by Parts · Level 3
Evaluate the integral: \(\displaystyle\int_{1}^{sqrt}(3) \arctan\left(\dfrac{1}{x}\right) d x\).
31 Exercise - Definite Integration by Parts · Level 2
Evaluate the integral: \(\displaystyle\int_{1}^{5} \dfrac{M}{e^M} d M\).
32 Exercise - Definite Integration by Parts · Level 2
Evaluate the integral: \(\displaystyle\int_{1}^{2} \dfrac{(\ln x)^2}{x^3} d x\).
33 Exercise - Definite Integration by Parts · Level 3
Evaluate the integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{3}} \sin x \ln(\cos x) d x\).
34 Exercise - Definite Integration by Parts · Level 3
Evaluate the integral: \(\displaystyle\int_{0}^{1} \dfrac{r^3}{\sqrt{4 + r^2}} d r\).
35 Exercise - Definite Integration by Parts · Level 3
Evaluate the integral: \(\displaystyle\int_{1}^{2} x^4 (\ln x)^2 d x\).
36 Exercise - Definite Integration by Parts · Level 3
Evaluate the integral: \(\displaystyle\int_{0}^{t} e^s \sin(t - s) d s\).
37 Exercise - Substitution Then By Parts · Level 3
First make a substitution and then use integration by parts to evaluate the integral: \(\int e^\sqrt{x} d x\).
38 Exercise - Substitution Then By Parts · Level 3
First make a substitution and then use integration by parts to evaluate the integral: \(\int \cos(\ln x) d x\).
39 Exercise - Substitution Then By Parts · Level 3
First make a substitution and then use integration by parts to evaluate the integral: \(\displaystyle\int_{\sqrt{\dfrac{\pi}{2}}}^\sqrt{\pi} \theta^3 \cos(\theta^2) d \theta\).
40 Exercise - Substitution Then By Parts · Level 3
First make a substitution and then use integration by parts to evaluate the integral: \(\displaystyle\int_{0}^{\pi} e^{\cos t} \sin 2t d t\).
41 Exercise - Substitution Then By Parts · Level 3
First make a substitution and then use integration by parts to evaluate the integral: \(\int x \ln(1 + x) d x\).
42 Exercise - Substitution Then By Parts · Level 3
First make a substitution and then use integration by parts to evaluate the integral: \(\int \dfrac{\arcsin(\ln x)}{x} d x\).
43 Exercise - Graphical Verification · Level 2
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take \(C = 0\)): \(\int x e^{-2x} d x\).
44 Exercise - Graphical Verification · Level 2
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take \(C = 0\)): \(\int x^{\dfrac{3}{2}} \ln x d x\).
45 Exercise - Graphical Verification · Level 3
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take \(C = 0\)): \(\int x^3 \sqrt{1 + x^2} d x\).
46 Exercise - Graphical Verification · Level 3
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take \(C = 0\)): \(\int x^2 \sin 2x d x\).
47 Exercise - Reduction Formula Application · Level 3
(a) Use the reduction formula in Example 6 to show that \(\int \sin^2 x d x = \dfrac{x}{2} - \dfrac{\sin 2x}{4} + C\). (b) Use part (a) and the reduction formula to evaluate \(\int \sin^4 x d x\).
48 Exercise - Reduction Formula · Level 3
(a) Prove the reduction formula \(\int \cos^n x d x = \dfrac{1}{n} \cos^{n-1} x \sin x + \dfrac{n-1}{n} \int \cos^{n-2} x d x\). (b) Use part (a) to evaluate \(\int \cos^2 x d x\). (c) Use parts (a) and (b) to evaluate \(\int \cos^4 x d x\).
49 Exercise - Reduction Formula Application · Level 3
(a) Use the reduction formula in Example 6 to show that \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin^n x d x = \dfrac{n-1}{n} \displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin^{n-2} x d x\) where \(n \geq 2\) is an integer. (b) Use part (a) to evaluate \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin^3 x d x\) and \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin^5 x d x\). (c) Use part (a) to show that, for odd powers of sine, \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin^{2n+1} x d x = \dfrac{2 \cdot 4 \cdot 6 \cdot \ldots \cdot 2n}{3 \cdot 5 \cdot 7 \cdot \ldots \cdot (2n+1)}\).
50 Exercise - Reduction Formula Proof · Level 3
Prove that, for even powers of sine, \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin^{2n} x d x = \dfrac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \ldots \cdot 2n} \cdot \dfrac{\pi}{2}\).
51 Exercise - Reduction Formula Proof · Level 3
Use integration by parts to prove the reduction formula: \(\int (\ln x)^n d x = x (\ln x)^n - n \int (\ln x)^{n-1} d x\).
52 Exercise - Reduction Formula Proof · Level 3
Use integration by parts to prove the reduction formula: \(\int x^n e^x d x = x^n e^x - n \int x^{n-1} e^x d x\).
53 Exercise - Reduction Formula Proof · Level 3
Use integration by parts to prove the reduction formula: \(\int \tan^n x d x = \dfrac{\tan^{n-1} x}{n-1} - \int \tan^{n-2} x d x\) \((n \neq 1)\).
54 Exercise - Reduction Formula Proof · Level 3
Use integration by parts to prove the reduction formula: \(\int \sec^n x d x = \dfrac{\tan x \sec^{n-2} x}{n-1} + \dfrac{n-2}{n-1} \int \sec^{n-2} x d x\) \((n \neq 1)\).
55 Exercise - Apply Reduction Formula · Level 2
Use Exercise 51 to find \(\int (\ln x)^3 d x\).
56 Exercise - Apply Reduction Formula · Level 2
Use Exercise 52 to find \(\int x^4 e^x d x\).
57 Exercise - Area Between Curves · Level 3
Find the area of the region bounded by the given curves: \(y = x^2 \ln x\), \(y = 4 \ln x\).
58 Exercise - Area Between Curves · Level 3
Find the area of the region bounded by the given curves: \(y = x^2 e^{-x}\), \(y = x e^{-x}\).
59 Exercise - Graphical Area · Level 3
Use a graph to find approximate \(x\)-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves: \(y = \arcsin\left(\dfrac{1}{2} x\right)\), \(y = 2 - x^2\).
60 Exercise - Graphical Area · Level 3
Use a graph to find approximate \(x\)-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves: \(y = x \ln(x + 1)\), \(y = 3x - x^2\).
61 Exercise - Volume by Cylindrical Shells · Level 3
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis: \(y = \cos\left(\pi \dfrac{x}{2}\right)\), \(y = 0\), \(0 \leq x \leq 1\); about the \(y\)-axis.
62 Exercise - Volume by Cylindrical Shells · Level 3
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis: \(y = e^x\), \(y = e^{-x}\), \(x = 1\); about the \(y\)-axis.
63 Exercise - Volume by Cylindrical Shells · Level 3
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis: \(y = e^{-x}\), \(y = 0\), \(x = -1\), \(x = 0\); about \(x = 1\).
64 Exercise - Volume by Cylindrical Shells · Level 3
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis: \(y = e^x\), \(x = 0\), \(y = 3\); about the \(x\)-axis.
65 Exercise - Volume Comparison · Level 3
Calculate the volume generated by rotating the region bounded by the curves \(y = \ln x\), \(y = 0\), and \(x = 2\) about each axis. (a) The \(y\)-axis. (b) The \(x\)-axis.
66 Exercise - Average Value · Level 2
Calculate the average value of \(f(x) = x \sec^2 x\) on the interval \([0, \dfrac{\pi}{4}]\).
67 Exercise - Fresnel Function · Level 3
The Fresnel function \(S(x) = \displaystyle\int_{0}^{x} \sin\left(\dfrac{1}{2} \pi t^2\right) d t\) was discussed in Example 5.3.3 and is used extensively in the theory of optics. Find \(\int S(x) d x\). [Your answer will involve \(S(x)\).]
68 Exercise - Rocket Application · Level 3
A rocket accelerates by burning its onboard fuel, so its mass decreases with time. Suppose the initial mass of the rocket at liftoff (including its fuel) is \(m\), the fuel is consumed at rate \(r\), and the exhaust gases are ejected with constant velocity \(v_e\) (relative to the rocket). A model for the velocity of the rocket at time \(t\) is given by the equation \(v(t) = -g t - v_e \ln \dfrac{m - r t}{m}\) where \(g\) is the acceleration due to gravity and \(t\) is not too large. If \(g = 9.8 m/s^2\), \(m = 30,000\) kg, \(r = 160\) kg/s, and \(v_e = 3000\) m/s, find the height of the rocket one minute after liftoff.
69 Exercise - Particle Motion · Level 2
A particle that moves along a straight line has velocity \(v(t) = t^2 e^{-t}\) meters per second after \(t\) seconds. How far will it travel during the first \(t\) seconds?
70 Exercise - Proof · Level 3
If \(f(0) = g(0) = 0\) and \(f''\) and \(g''\) are continuous, show that \(\displaystyle\int_{0}^{a} f(x) g''(x) d x = f(a) g'(a) - f'(a) g(a) + \displaystyle\int_{0}^{a} f''(x) g(x) d x\).
71 Exercise - Integration by Parts Application · Level 3
Suppose that \(f(1) = 2\), \(f(4) = 7\), \(f'(1) = 5\), \(f'(4) = 3\), and \(f''\) is continuous. Find the value of \(\displaystyle\int_{1}^{4} x f''(x) d x\).
72 Exercise - Inverse Functions Theory · Level 3
(a) Use integration by parts to show that \(\int f(x) d x = x f(x) - \int x f'(x) d x\). (b) If \(f\) and \(g\) are inverse functions and \(f'\) is continuous, prove that \(\displaystyle\int_{a}^{b} f(x) d x = b f(b) - a f(a) - \displaystyle\int_{f(a})^{f(b)} g(y) d y\). [Hint: Use part (a) and make the substitution \(y = f(x)\).] (c) In the case where \(f\) and \(g\) are positive functions and \(b > a > 0\), draw a diagram to give a geometric interpretation of part (b). (d) Use part (b) to evaluate \(\displaystyle\int_{1}^{e} \ln x d x\).
73 Exercise - Volume Theory · Level 3
We arrived at Formula 6.3.2, \(V = \displaystyle\int_{a}^{b} 2 \pi x f(x) d x\), by using cylindrical shells, but now we can use integration by parts to prove it using the slicing method of Section 6.2, at least for the case where \(f\) is one-to-one and therefore has an inverse function \(g\). Use the figure to show that \(V = \pi b^2 d - \pi a^2 c - \displaystyle\int_{c}^{d} \pi [g(y)]^2 d y\). Make the substitution \(y = f(x)\) and then use integration by parts on the resulting integral to prove that \(V = \displaystyle\int_{a}^{b} 2 \pi x f(x) d x\).
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74 Exercise - Wallis Product · Level 3
Let \(I_n = \displaystyle\int_{0}^{\dfrac{\pi}{2}} \sin^n x d x\). (a) Show that \(I_{2n+2} \leq I_{2n+1} \leq I_{2n}\). (b) Use Exercise 50 to show that \(\dfrac{I_{2n+2}}{I_{2n}} = \dfrac{2n+1}{2n+2}\). (c) Use parts (a) and (b) to show that \(\dfrac{2n+1}{2n+2} \leq \dfrac{I_{2n+1}}{I_{2n}} \leq 1\) and deduce that \(\operatorname*{lim}\limits_{n \rightarrow \infty} I_{2n+1}/I_{2n} = 1\). (d) Use part (c) and Exercises 49 and 50 to show that \(\operatorname*{lim}\limits_{n \rightarrow \infty} \dfrac{2}{1} \cdot \dfrac{2}{3} \cdot \dfrac{4}{3} \cdot \dfrac{4}{5} \cdot \dfrac{6}{5} \cdot \dfrac{6}{7} \cdot \ldots \cdot \dfrac{2n}{2n-1} \cdot \dfrac{2n}{2n+1} = \dfrac{\pi}{2}\). This is the Wallis product. (e) We construct rectangles as follows. Start with a square of area 1 and attach rectangles of area 1 alternately beside or on top of the previous rectangle. Find the limit of the ratios of width to height of these rectangles.
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75 Example - Integration by Parts · Level 2
Find \(\int x \sin x d x\).
76 Example - Integration by Parts · Level 2
Evaluate \(\int \ln x d x\).
77 Example - Integration by Parts (Twice) · Level 3
Find \(\int t^2 e^t d t\).
78 Example - Integration by Parts (Loop) · Level 3
Evaluate \(\int e^x \sin x d x\).
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79 Example - Definite Integration by Parts · Level 3
Calculate \(\displaystyle\int_{0}^{1} \tan^{-1} x d x\).
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80 Example - Reduction Formula · Level 3
Prove the reduction formula \(\int \sin^n x d x = -\dfrac{1}{n} \cos x \sin^{n-1} x + \dfrac{n-1}{n} \int \sin^{n-2} x d x\) where \(n \geq 2\) is an integer.

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