Calculus: Early Transcendentals
James Stewart
Educators
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 $
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 $
Evaluate the integral.
$ \displaystyle \int (x - 1) \sin \pi x dx $
Evaluate the integral.
$ \displaystyle \int (x^2 + 2x) \cos x dx $
Evaluate the integral.
$ \displaystyle \int t^2 \sin \beta t dt $
Evaluate the integral.
$ \displaystyle \int e^{2 \theta} \sin 3 \theta d \theta $
Evaluate the integral.
$ \displaystyle \int e^{-\theta} \cos 2 \theta d \theta $
Evaluate the integral.
$ \displaystyle \int \frac{xe^{2x}}{(1 + 2x)^2} dx $
Evaluate the integral.
$ \displaystyle \int_0^{\frac{1}{2}} x \cos \pi x dx $
Evaluate the integral.
$ \displaystyle \int_0^1 (x^2 + 1) e^{-x} dx $
Evaluate the integral.
$ \displaystyle \int_0^2 y \sinh y dy $
Evaluate the integral.
$ \displaystyle \int_1^2 w^2 \ln w dw $
Evaluate the integral.
$ \displaystyle \int_1^5 \frac{\ln R}{R^2} dR $
Evaluate the integral.
$ \displaystyle \int_0^{2 \pi} t^2 \sin 2t dt $
Evaluate the integral.
$ \displaystyle \int_0^\pi x \sin x \cos x dx $
Evaluate the integral.
$ \displaystyle \int_1^{\sqrt{3}} \arctan (\frac{1}{x}) dx $
Evaluate the integral.
$ \displaystyle \int_1^5 \frac{M}{e^M} dM $
Evaluate the integral.
$ \displaystyle \int_1^2 \frac{(\ln x)^2}{x^3} dx $
Evaluate the integral.
$ \displaystyle \int_0^{\frac{\pi}{3}} \sin x \ln (\cos x) dx $
Evaluate the integral.
$ \displaystyle \int_0^1 \frac{r^3}{\sqrt{4 + r^2}} dr $
Evaluate the integral.
$ \displaystyle \int_1^2 x^4 (\ln x)^2 dx $
Evaluate the integral.
$ \displaystyle \int_0^t e^s \sin (t - s) ds $
First make a substitution and then use integration by parts to evaluate the integral.
$ \displaystyle \int e^{\sqrt{x}} dx $
First make a substitution and then use integration by parts to evaluate the integral.
$ \displaystyle \int \cos (\ln x) dx $
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 $
First make a substitution and then use integration by parts to evaluate the integral.
$ \displaystyle \int_0^\pi e^{\cos t} \sin 2t dt $
First make a substitution and then use integration by parts to evaluate the integral.
$ \displaystyle \int x \ln (1 + x) dx $
First make a substitution and then use integration by parts to evaluate the integral.
$ \displaystyle \int \frac{\arcsin (\ln x)}{x} dx $
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 $
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 $
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 $
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 $
(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 $.
(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 $.
(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,
$$ \int_0^{\frac{\pi}{2}} \sin^{2n + 1} x dx = \frac{2 \cdot 4 \cdot 6 \cdots \cdots 2n}{3 \cdot 5 \cdot 7 \cdots \cdots (2n +1)} $$
Prove that, for even powers of sine,
$$ \int_0^{\frac{\pi}{2}} \sin^{2n} x dx = \frac{1 \cdot 3 \cdot 5 \cdots \cdots (2n - 1)}{2 \cdot 4 \cdot 6 \cdots \cdots 2n} \frac{\pi}{2} $$
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 $
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 $
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) $
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) $
Find the area of the region bounded by the given curves.
$ y = x^2 \ln x $ , $ y = 4 \ln x $
Find the area of the region bounded by the given curves.
$ y = x^2 e^{-x} $ , $ y = xe^{-x} $
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 $
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 $
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
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
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 $
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
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
Calculate the average value of $ f(x) = x \sec^2 x $ on the interval $ [0, \frac{\pi}{4}] $.
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) $.]
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.
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?
If $ f(0) = g(0) = 0 $ and $ f^n $ and $ g^n $ are continuous, show that
$$ \int_0^a f(x) g^{\prime\prime} (x) dx = f(a)g^\prime(a) - f^\prime(a)g(a) + \int_0^a f^{\prime\prime} (x) g(x) dx $$
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 $.
(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 $.
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
$$ V = \int_a^b 2 \pi x f(x) dx $$
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.
