Chapter 5

Evaluate the indefinite integral. \( \displaystyle \int \frac{\sin x}{1 + \cos^2 x} \, dx \)

Evaluate the integral. \( \displaystyle \int^{10}_{-10} \frac{2e^x}{\sinh x + \cosh x} \,dx \)

Evaluate the integral. \( \displaystyle \int^3_1 \frac{y^3 - 2y^2 - y}{y^2} \,dy \)

Evaluate the integral by interpreting it in terms of areas. \( \displaystyle \int^1_0 \bigl| 2x - 1 \bigr| \, dx \)

Evaluate the indefinite integral. \( \displaystyle \int \cot x \, dx \)

Evaluate the integral. \( \displaystyle \int^{\sqrt{3}/2}_{0} \frac{dr}{\sqrt{1 - r^2}} \)

Evaluate the integral. \( \displaystyle \int^4_0 2^s \,ds \)

Evaluate \( \displaystyle \int^1_1 \sqrt{1 + x^4}\, dx \).

Evaluate the indefinite integral. \( \displaystyle \int \frac{\cos (\ln t)}{t} \, dt \)

Evaluate the integral. \( \displaystyle \int^{2}_{1} \frac{(x - 1)^3}{x^2} \,dx \)

Evaluate the integral. \( \displaystyle \int^{1/\sqrt{2}}_{1/2} \frac{4}{\sqrt{1 - x^2}} \,dx \)

Given that \( \displaystyle \int^{\pi}_0 \sin ^4x \, dx = \frac{3}{8} \pi \), what is \( \displaystyle \int^0_{\pi} \sin ^4\theta \, d\theta \)?

Evaluate the indefinite integral. \( \displaystyle \int \frac{dx}{\sqrt{1 - x^2} \sin^{-1} x} \)

Evaluate the integral. \( \displaystyle \int^{1/\sqrt{3}}_{0} \frac{t^2 - 1}{t^4 - 1} \,dt \)

Evaluate the integral. \( \displaystyle \int^{\pi}_0 f(x) \,dx \) where \( f(x) = \left\\{ \begin{array}{ll} \sin x & \mbox{if \) 0 \le x < \pi/2 \(}\\\ \cos x & \mbox{if \) \pi/2 \le x \le \pi \(} \end{array} \right.\)

In Example 5.1.2 we showed that \( \displaystyle \int^1_0 x^2 \, dx = \frac{1}{3} \). Use this fact and the properties of integrals to evaluate \( \displaystyle \int^1_0 (5 - 6x^2) \, dx \).

Evaluate the indefinite integral. \( \displaystyle \int \frac{x}{1 + x^4} \, dx \)

Evaluate the integral. \( \displaystyle \int^{2}_{0} \mid 2x - 1 \mid \,dx \)

Evaluate the integral. \( \displaystyle \int^2_{-2} f(x) \,dx \) where \( f(x) = \left\\{ \begin{array}{ll} 2 & \mbox{if \) -2 \le x \le 0 \(}\\\ 4 - x^2 & \mbox{if \) 0 < x \le 2 \(} \end{array} \right.\)

Evaluate the indefinite integral. \( \displaystyle \int \frac{1 + x}{1 + x^2} \, dx \)

Evaluate the integral. \( \displaystyle \int^{2}_{-1} \bigl( x - 2 \mid x \mid \bigr) \,dx \)

Sketch the region enclosed by the given curves and calculate its area. \( y = \sqrt{x} \), \( y = 0 \), \( x = 4 \)

Evaluate the indefinite integral. \( \displaystyle \int x^2 \sqrt{2 + x} \, dx \)

Evaluate the integral. \( \displaystyle \int^{3\pi/2}_{0} \mid \sin x \mid \,dx \)

Sketch the region enclosed by the given curves and calculate its area. \( y = x^3 \), \( y = 0 \), \( x = 1 \)

Evaluate the indefinite integral. \( \displaystyle \int x(2x + 5)^8 \, dx \)

Sketch the region enclosed by the given curves and calculate its area. \( y = 4 - x^2 \), \( y = 0 \)

Write as a single integral in the form \( \displaystyle \int^b_a f(x) \, dx \): $$ \int^2_{-2} f(x) \, dx + \int^5_2 f(x) \, dx - \int^{-1}_{-2} f(x) \, dx $$

Evaluate the indefinite integral. \( \displaystyle \int x^3 \sqrt{x^2 + 1} \, dx \)

Sketch the region enclosed by the given curves and calculate its area. \( y = 2x - x^2 \), \( y = 0 \)

If \( \displaystyle \int^8_2 f(x) \, dx = 7.3 \) and \( \displaystyle \int^4_2 f(x) \, dx = 5.9 \), find \( \displaystyle \int^8_4 f(x) \, dx \).

Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take \( C = 0 \)). \( \displaystyle \int x(x^2 - 1)^3 \, dx \)

The area of the region that lies to the right of the \( y \)-axis and to the left of the parabola \( x = 2y - y^2 \) (the shaded region in the figure) is given by the integral \( \displaystyle \int^2_0 (2y - y^2) \, dy \). (Turn your head clockwise and think of the region as lying below the curve \( x = 2y - y^2 \) from \( y = 0 \) to \( y = 2 \).) Find the area of the region.

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. \( y = \sqrt[3]{x} \), \( 0 \le x \le 27 \)

If \( \displaystyle \int^9_0 f(x) \, dx = 37 \) and \( \displaystyle \int^9_0 g(x) \, dx = 16 \), find $$ \int^9_0 \bigl[ 2f(x) + 3g(x) \bigr] \, dx $$

Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take \( C = 0 \)). \( \displaystyle \int \tan^2 \theta \sec^2 \theta \, d\theta \)

The boundaries of the shaded region are the \( y \)-axis, the line \( y = 1 \), and the curve \( y = \sqrt[4]{x} \). Find the area of this region by writing \( x \) as a function of \( y \) and integrating with respect to \( y \) (as in Exercise 49).

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. \( y = x^{-4} \), \( 1 \le x \le 6 \)

Find \( \displaystyle \int^5_0 f(x) \, dx \) if \( f(x) = \left\\{ \begin{array}{ll} 3 & \mbox{if \) x < 3 \(}\\\ x & \mbox{if \) x \ge 3 \(} \end{array} \right.\)

Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take \( C = 0 \)). \( \displaystyle \int e^{\cos x} \sin x \, dx \)

If \( w'(t) \) is the rate of growth of a child in pounds per year, what does\( \displaystyle \int^{10}_5 w'(t) \,dt \) represent?

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. \( y = \sin x \), \( 0 \le x \le \pi \)

Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take \( C = 0 \)). \( \displaystyle \int \sin x \cos^4 x \, dx \)

The current in a wire is defined as the derivative of the charge: \( I(t) = Q'(t) \). (See Example 3.7.3.) What does \( \displaystyle \int^b_a I(t) \, dt \) represent?

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. \( y = \sec^2 x \), \( 0 \le x \le \pi/3 \)

Evaluate the definite integral. \( \displaystyle \int^1_0 \cos(\pi t/2) \, dt \)

If oil leaks from a tank at a rate of \( r(t) \) gallons per minute at time \( t \), what does \( \displaystyle \int^{120}_0 r(t) \, dt \) represent?

Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch. \( \displaystyle \int^2_{-1} x^3 \,dx \)

Evaluate the definite integral. \( \displaystyle \int^1_0 (3t - 1)^{50} \, dt \)

A honeybee population starts with 100 bees and increases at a rate of \( n'(t) \) bees per week. What does \( \displaystyle 100 + \int^{15}_0 n'(t) \, dt \) represent?