Ask your homework questions to teachers and professors, meet other students, and be entered to win $600 or an Xbox Series X 🎉Join our Discord! ## Chapter 7 ## Techniques of Integration ## Educators SL DQ WZ WJ ### Problem 1 Evaluate the integral using integration by parts with the indicated choices of$ u $and$ dv $.$ \displaystyle \int xe^{2x} $;$ u = x $,$ dv = e^{2x} dx $SL Sky L. Numerade Educator ### Problem 2 Evaluate the integral using integration by parts with the indicated choices of$ u $and$ dv $.$ \displaystyle \int \sqrt{x} \ln x dx $;$ u = \displaystyle \ln x $,$ dv = \sqrt{x} dx $DQ Danjoseph Q. Numerade Educator ### Problem 3 Evaluate the integral.$ \displaystyle \int x \cos 5x dx $DQ Danjoseph Q. Numerade Educator ### Problem 4 Evaluate the integral.$ \displaystyle \int ye^{0.2y} dy $SL Sky L. Numerade Educator ### Problem 5 Evaluate the integral.$ \displaystyle \int te^{-3t} dt $DQ Danjoseph Q. Numerade Educator ### Problem 6 Evaluate the integral.$ \displaystyle \int (x - 1) \sin \pi x dx $DQ Danjoseph Q. Numerade Educator ### Problem 7 Evaluate the integral.$ \displaystyle \int (x^2 + 2x) \cos x dx $WZ Wen Z. Numerade Educator ### Problem 8 Evaluate the integral.$ \displaystyle \int t^2 \sin \beta t dt $Carson M. Numerade Educator ### Problem 9 Evaluate the integral.$ \displaystyle \int \cos^{-1} x dx $WZ Wen Z. Numerade Educator ### Problem 10 Evaluate the integral.$ \displaystyle \int \ln \sqrt{x} dx $WZ Wen Z. Numerade Educator ### Problem 11 Evaluate the integral.$ \displaystyle \int t^4 \ln t dt $WZ Wen Z. Numerade Educator ### Problem 12 Evaluate the integral.$ \displaystyle \int \tan^{-1} 2y dy $WZ Wen Z. Numerade Educator ### Problem 13 Evaluate the integral.$ \displaystyle \int t \csc^2 t dt $WZ Wen Z. Numerade Educator ### Problem 14 Evaluate the integral.$ \displaystyle \int x \cosh ax dx $WZ Wen Z. Numerade Educator ### Problem 15 Evaluate the integral.$\int \frac{x-1}{x^{2}+2 x} d x$WZ Wen Z. Numerade Educator ### Problem 16 Evaluate the integral.$ \displaystyle \int \frac{z}{10^z} dz $WZ Wen Z. Numerade Educator ### Problem 17 Evaluate the integral.$ \displaystyle \int e^{2 \theta} \sin 3 \theta d \theta $WZ Wen Z. Numerade Educator ### Problem 18 Evaluate the integral.$ \displaystyle \int e^{-\theta} \cos 2 \theta d \theta $WZ Wen Z. Numerade Educator ### Problem 19 Evaluate the integral.$ \displaystyle \int z^3 e^z dz $WZ Wen Z. Numerade Educator ### Problem 20 Evaluate the integral.$ \displaystyle \int x \tan^2 x dx $WZ Wen Z. Numerade Educator ### Problem 21 Evaluate the integral.$ \displaystyle \int \frac{xe^{2x}}{(1 + 2x)^2} dx $WZ Wen Z. Numerade Educator ### Problem 22 Evaluate the integral.$ \displaystyle \int (\arcsin x)^2 dx $WZ Wen Z. Numerade Educator ### Problem 23 Evaluate the integral.$ \displaystyle \int_0^{\frac{1}{2}} x \cos \pi x dx $WZ Wen Z. Numerade Educator ### Problem 24 Evaluate the integral.$ \displaystyle \int_0^1 (x^2 + 1) e^{-x} dx $WZ Wen Z. Numerade Educator ### Problem 25 Evaluate the integral.$ \displaystyle \int_0^2 y \sinh y dy $WJ Willis J. Numerade Educator ### Problem 26 Evaluate the integral.$ \displaystyle \int_1^2 w^2 \ln w dw $WJ Willis J. Numerade Educator ### Problem 27 Evaluate the integral.$ \displaystyle \int_1^5 \frac{\ln R}{R^2} dR $WZ Wen Z. Numerade Educator ### Problem 28 Evaluate the integral.$ \displaystyle \int_0^{2 \pi} t^2 \sin 2t dt $WZ Wen Z. Numerade Educator ### Problem 29 Evaluate the integral.$ \displaystyle \int_0^\pi x \sin x \cos x dx $WZ Wen Z. Numerade Educator ### Problem 30 Evaluate the integral.$ \displaystyle \int_1^{\sqrt{3}} \arctan (\frac{1}{x}) dx $WZ Wen Z. Numerade Educator ### Problem 31 Evaluate the integral.$ \displaystyle \int_1^5 \frac{M}{e^M} dM $WZ Wen Z. Numerade Educator ### Problem 32 Evaluate the integral.$ \displaystyle \int_1^2 \frac{(\ln x)^2}{x^3} dx $WZ Wen Z. Numerade Educator ### Problem 33 Evaluate the integral.$ \displaystyle \int_0^{\frac{\pi}{3}} \sin x \ln (\cos x) dx $WZ Wen Z. Numerade Educator ### Problem 34 Evaluate the integral.$ \displaystyle \int_0^1 \frac{r^3}{\sqrt{4 + r^2}} dr $WZ Wen Z. Numerade Educator ### Problem 35 Evaluate the integral.$ \displaystyle \int_1^2 x^4 (\ln x)^2 dx $WZ Wen Z. Numerade Educator ### Problem 36 Evaluate the integral.$ \displaystyle \int_0^t e^s \sin (t - s) ds $WZ Wen Z. Numerade Educator ### Problem 37 First make a substitution and then use integration by parts to evaluate the integral.$ \displaystyle \int e^{\sqrt{x}} dx $WZ Wen Z. Numerade Educator ### Problem 38 First make a substitution and then use integration by parts to evaluate the integral.$ \displaystyle \int \cos (\ln x) dx $WZ Wen Z. Numerade Educator ### Problem 39 First make a substitution and then use integration by parts to evaluate the integral.$ \displaystyle \int_{\sqrt{\frac{\pi}{2}}}^{\sqrt{\pi}} \theta^3 \cos (\theta^2) d \theta $WZ Wen Z. Numerade Educator ### Problem 40 First make a substitution and then use integration by parts to evaluate the integral.$ \displaystyle \int_0^\pi e^{\cos t} \sin 2t dt $WZ Wen Z. Numerade Educator ### Problem 41 First make a substitution and then use integration by parts to evaluate the integral.$ \displaystyle \int x \ln (1 + x) dx $WZ Wen Z. Numerade Educator ### Problem 42 First make a substitution and then use integration by parts to evaluate the integral.$ \displaystyle \int \frac{\arcsin (\ln x)}{x} dx $WZ Wen Z. Numerade Educator ### Problem 43 Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take$ C = 0 $).$ \displaystyle \int xe^{-2x} dx $WZ Wen Z. Numerade Educator ### Problem 44 Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take$ C = 0 $).$ \displaystyle \int x^{\frac{3}{2}} \ln x dx $WZ Wen Z. Numerade Educator ### Problem 45 Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take$ C = 0 $).$ \displaystyle \int x^3 \sqrt{1 + x^2} dx $WZ Wen Z. Numerade Educator ### Problem 46 Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take$ C = 0 $).$ \displaystyle \int x^2 \sin 2x dx $WZ Wen Z. Numerade Educator ### Problem 47 (a) Use the reduction formula in Example 6 to show that $$\int \sin^2 x dx = \frac{x}{2} - \frac{\sin 2x}{4} + C$$ (b) Use part (a) and the reduction formula to evaluate$ \displaystyle \int \sin^4 x dx $. WJ Willis J. Numerade Educator ### Problem 48 (a) Prove the reduction formula $$\int \cos^n x dx = \frac{1}{n} \cos^{n - 1} x \sin x + \frac{n - 1}{n} \int \cos^{n - 2} x dx$$ (b) Use part (a) to evaluate$ \displaystyle \int \cos^2 x dx $. (c) Use parts (a) and (b) to evaluate$ \displaystyle \int \cos^4 x dx $. Carson M. Numerade Educator ### Problem 49 (a) Use the reduction formula in Example 6 to show that $$\int_0^{\frac{\pi}{2}} \sin^n x dx = \frac{n - 1}{n} \int_0^{\frac{\pi}{2}} \sin^{n - 2} x dx$$ where$ n \ge 2 $is an integer. (b) Use part (a) to evaluate$ \displaystyle \int_0^{\frac{\pi}{2}} \sin^3 x dx $and$ \displaystyle \int_0^{\frac{\pi}{2}} \sin^5 x dx $. (c) Use part (a) to show that, for odd powers of sine, $$\int_0^{\frac{\pi}{2}} \sin^{2n + 1} x dx = \frac{2 \cdot 4 \cdot 6 \cdots \cdots 2n}{3 \cdot 5 \cdot 7 \cdots \cdots (2n +1)}$$ WJ Willis J. Numerade Educator ### Problem 50 Prove that, for even powers of sine, $$\int_0^{\frac{\pi}{2}} \sin^{2n} x dx = \frac{1 \cdot 3 \cdot 5 \cdots \cdots (2n - 1)}{2 \cdot 4 \cdot 6 \cdots \cdots 2n} \frac{\pi}{2}$$ WZ Wen Z. Numerade Educator ### Problem 51 Use integration by parts to prove the reduction formula.$ \displaystyle \int (\ln x)^n dx = x (\ln x)^n - n \displaystyle \int (\ln x)^{n - 1} dx $WZ Wen Z. Numerade Educator ### Problem 52 Use integration by parts to prove the reduction formula.$ \displaystyle \int x^n e^x dx = x^n e^x - n \displaystyle \int x^{n - 1} e^x dx $WZ Wen Z. Numerade Educator ### Problem 53 Use integration by parts to prove the reduction formula.$ \displaystyle \int \tan^n x dx = \frac{\tan^{n -1} x}{n - 1} - \displaystyle \int \tan^{n -2} x dx (n \neq 1) $WZ Wen Z. Numerade Educator ### Problem 54 Use integration by parts to prove the reduction formula.$ \displaystyle \int \sec^n x dx = \frac{\tan x \sec^{n - 2} x}{n - 1} + \frac{n - 2}{n - 1} \displaystyle \int \sec^{n - 2} x dx (n \neq 1) $WZ Wen Z. Numerade Educator ### Problem 55 Use Exercise 51 to find$ \displaystyle \int (\ln x)^3 dx $. WZ Wen Z. Numerade Educator ### Problem 56 Use Exercise 52 to find$ \displaystyle \int x^4 e^x dx $. WZ Wen Z. Numerade Educator ### Problem 57 Find the area of the region bounded by the given curves.$ y = x^2 \ln x $,$ y = 4 \ln x $WZ Wen Z. Numerade Educator ### Problem 58 Find the area of the region bounded by the given curves.$ y = x^2 e^{-x} $,$ y = xe^{-x} $WZ Wen Z. Numerade Educator ### Problem 59 Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.$ y = \arcsin \left(\frac{1}{2} x \right) $,$ y = 2 - x^2 $WZ Wen Z. Numerade Educator ### Problem 60 Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.$ y = x \ln (x + 1) $,$ y = 3x - x^2 $WZ Wen Z. Numerade Educator ### Problem 61 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.$ y = \cos (\frac{\pi x}{2}) $,$ y = 0 $,$ 0 \le x \le 1 $; about the y-axis WZ Wen Z. Numerade Educator ### Problem 62 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.$ y = e^x $,$ y = e^{-x} $,$ x = 1 $; about the y-axis WZ Wen Z. Numerade Educator ### Problem 63 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.$ y = e^{-x} $,$ y = 0 $,$ x = -1 $,$ x = 0 $; about$ x = 1 $WZ Wen Z. Numerade Educator ### Problem 64 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.$ y = e^x $,$ x = 0 $,$ y = 3 $; about the x-axis WZ Wen Z. Numerade Educator ### Problem 65 Calculate the volume generated by rotating the region bounded by the curves$ y = \ln x $,$ y = 0 $and$ x = 2 $about each axis. (a) The y-axis (b) The x-axis WZ Wen Z. Numerade Educator ### Problem 66 Calculate the average value of$ f(x) = x \sec^2 x $on the interval$ [0, \frac{\pi}{4}] $. WZ Wen Z. Numerade Educator ### Problem 67 The Fresnel function$ S(x) = \displaystyle \int_0^x \sin \left(\frac{1}{2} \pi t^2 \right) dt $was discussed in Example 5.3.3 and is used extensively in the theory of optics. Find$ S(x) dx $. [Your answer will involve$ S(x) $.] Carson M. Numerade Educator ### Problem 68 A rocket accelerates by burning its onboard fuel, so its mass decreases with time. Suppose the initial mass of the rocket at liftoff (including its fuel) is$ 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 $$v(t) = -gt - v_e \ln \frac{m - rt}{m}$$ 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 $, find the height of the rocket one minute after liftoff. Carson M. Numerade Educator ### Problem 69 A particle that moves along a straight line has velocity$ v(t) = t^2 e^{-t} $meters per second after$ t $seconds. How far will it travel during the first$ t $seconds? WZ Wen Z. Numerade Educator ### Problem 70 If$ f(0) = g(0) = 0 $and$ f^n $and$ g^n $are continuous, show that $$\int_0^a f(x) g^{\prime\prime} (x) dx = f(a)g^\prime(a) - f^\prime(a)g(a) + \int_0^a f^{\prime\prime} (x) g(x) dx$$ WZ Wen Z. Numerade Educator ### Problem 71 Suppose that$ f(1) = 2 $,$ f(4) = 7 $,$ f^\prime(1) = 5 $,$ f^\prime(4) = 3 $and$ f^{\prime\prime} $is continuous. Find the value of$ \displaystyle \int_1^4 x f^{\prime\prime} (x)\ dx $. WJ Willis J. Numerade Educator ### Problem 72 (a) Use integration by parts to show that $$\int f(x) dx = xf (x) - \int xf^\prime (x) dx$$ (b) If$ f $and$ g $are inverse functions and$ f^\prime $is continuous, prove that $$\int_a^b f(x) dx = bf (b) - af (a) - \int_{f(a)}^{f(b)} g(y) dy$$ [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 dx $. WZ Wen Z. Numerade Educator ### Problem 73 We arrived at Formula 6.3.2,$ V = \displaystyle \int_a^b 2 \pi x f(x) dx $, by using cylindrical shells, but now we can use integration by parts to prove it using the slicing method of Section 6.2, at least for the case where$ f $is one-to-one and therefore has an inverse function$ g $. Use the figure to show that $$V = \pi b^2 d - \pi a^2 c - \int_c^d \pi [g(y)]^2 dy$$ Make the substitution$ y = f(x) $and then use integration by parts on the resulting integral to prove that $$V = \int_a^b 2 \pi x f(x) dx$$ WZ Wen Z. Numerade Educator ### Problem 74 Let$ I_n = \displaystyle \int_0^{\frac{\pi}{2}} \sin^n x dx $. (a) Show that$ I_{2n + 2} \le I_{2n + 1} \le I_{2n} $. (b) Use Exercise 50 to show that $$\frac{I_{2n + 2}}{I_{2n}} = \frac{2n + 1}{2n + 2}$$ (c) Use parts (a) and (b) to show that $$\frac{2n + 1}{2n + 2} \le \frac{I_{2n + 1}}{I_{2n}} \le 1$$ and deduce that$ \lim_{n\to\infty}\frac{I_{2n + 1}}{I_{2n}} = 1 \$.
(d) Use part (c) and Exercises 49 and 50 to show that
$$\lim_{n\to\infty} \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots \cdots \frac{2n}{2n - 1} \cdot \frac{2n}{2n + 1} = \frac{\pi}{2}$$
This formula is usually written as an infinite product:
$$\frac{\pi}{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots$$
and is called 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 (see the figure). Find the limit of the ratios of width to height of these rectangles.

WZ
Wen Z.