Chapter 5

Evaluate the integral. \( \displaystyle \int^1_0 (u + 2) (u - 3) \,du \)

Prove that \( \displaystyle \int^b_a x \, dx = \frac{b^2 - a^2}{2} \).

Let \( A \) be the area under the graph of an increasing continuous function \( f \) from \( a \) to \( b \), and let \( L_n \) and \( R_n \) be the approximations to \( A \) with \( n \) subintervals using left and right endpoints, respectively. (a) How are \( A \), \( L_n \), and \( R_n \) related? (b) Show that $$ R_n - L_n = \frac{b - a}{n} [f(b) - f(a)] $$ Then draw a diagram to illustrate this equation by showing that the \( n \) rectangles representing \( R_n - L_n \) can be reassembled to form a single rectangle whose area is the right side of the equation. (c) Deduce that $$ R_n - A < \frac{b - a}{n} [f(b) - f(a)] $$

Evaluate the indefinite integral. \( \displaystyle \int e^{\cos t} \sin t \, dt \)

Evaluate the integral. \( \displaystyle \int^{2}_{1} \biggl( \frac{1}{x^2} - \frac{4}{x^3} \biggr) \,dx \)

Evaluate the integral. \(\int_{0}^{4}(4-t) \sqrt{t} d t\)

Prove that \( \displaystyle \int^b_a x^2 \, dx = \frac{b^3 - a^3}{3} \).

Evaluate the indefinite integral. \( \displaystyle \int 5^t \sin(5^t) \, dt \)

Evaluate the integral. \( \displaystyle \int^{4}_{1} \biggl( \frac{4 + 6u}{\sqrt{u}} \biggr) \,du \)

Evaluate the integral. \( \displaystyle \int^4_1 \frac{2 + x^2}{\sqrt{x}} \,dx \)

Express the integral as a limit of Riemann sums. Do not evaluate the limit. \( \displaystyle \int^3_1 \sqrt{4 + x^2} \, dx \)

(a) Express the area under the curve \( y = x^5 \) from 0 to 2 as a limit. (b) Use a computer algebra system to find the sum in your expression from part (a). (c) Evaluate the limit in part (a).

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

Evaluate the integral. \( \displaystyle \int^{1}_{0} \frac{4}{1 + p^2} \,dp \)

Evaluate the integral. \( \displaystyle \int^2_{-1} (3u - 2)(u + 1) \,du \)

Express the integral as a limit of Riemann sums. Do not evaluate the limit. \( \displaystyle \int^5_2 \biggl( x^2 + \frac{1}{x} \biggr) \, dx \)

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

Evaluate the integral. \( \displaystyle \int^{1}_{0} x \bigl( \sqrt[3]{x} + \sqrt[4]{x} \bigr) \,dx \)

Evaluate the integral. \( \displaystyle \int^{\pi/2}_{\pi/6} \csc t \cot t \,dt \)

Express the integral as a limit of sums. Then evaluate, using a computer algebra system to find both the sum and the limit. \( \displaystyle \int^{\pi}_0 \sin 5x \, dx \)

Find the exact area under the cosine curve \( y = \cos x \) from \( x = 0 \) to \( x = b \), where \( 0 \le b \le \pi/2 \). (Use a computer algebra system both to evaluate the sum and compute the limit.) In particular, what is the area if \( b = \pi/2 \)?

Evaluate the indefinite integral. \( \displaystyle \int \frac{x}{x^2 + 4} \, dx \)

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

Evaluate the integral. \( \displaystyle \int^{\pi/3}_{\pi/4} \csc^2 \theta \,d\theta \)

Express the integral as a limit of sums. Then evaluate, using a computer algebra system to find both the sum and the limit. \( \displaystyle \int^{10}_2 x^6 \, dx \)

Evaluate the indefinite integral. \( \displaystyle \int \cos (1 + 5t) \, dt \)

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

Evaluate the integral. \( \displaystyle \int^1_0 (1 + r)^3 \,dr \)

Evaluate the indefinite integral. \( \displaystyle \int \frac{\cos (\pi/x)}{x^2} \, dx \)

Evaluate the integral. \( \displaystyle \int^{1}_{0} (5x - 5^x) \,dx \)

Evaluate the integral. \( \displaystyle \int^3_0 (2\sin x - e^x) \,dx \)

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

Evaluate the integral. \( \displaystyle \int^{1}_{0} (x^{10} + 10^x)\,dx \)

Evaluate the integral. \( \displaystyle \int^2_1 \frac{v^3 + 3v^6}{v^4} \,dv \)

Evaluate the integral by interpreting it in terms of areas. \( \displaystyle \int^2_{-1} (1 - x) \, dx \)

Evaluate the indefinite integral. \( \displaystyle \int \frac{2^t}{2^t + 3} \, dt \)

Evaluate the integral. \( \displaystyle \int^{\pi/4}_{0} \sec \theta \tan \theta \,d\theta \)

Evaluate the integral. \( \displaystyle \int^{18}_1 \sqrt{\frac{3}{z}} \,dz \)

Evaluate the integral by interpreting it in terms of areas. \( \displaystyle \int^9_0 \biggl( \frac{1}{3}x - 2 \biggr) \, dx \)

Evaluate the indefinite integral. \( \displaystyle \int \sinh^2 x \cosh x \, dx \)

Evaluate the integral. \( \displaystyle \int^{\pi/4}_{0} \frac{1 + \cos^2 \theta}{\cos^2 \theta} \,d\theta \)

Evaluate the integral by interpreting it in terms of areas. \( \displaystyle \int^0_{-3} \bigl( 1 + \sqrt{9 - x^2} \bigr) \, dx \)

Evaluate the indefinite integral. \( \displaystyle \int \frac{dt}{\cos^2 t \sqrt{1 + \tan t}} \)

Evaluate the integral. \( \displaystyle \int^{\pi/3}_{0} \frac{\sin \theta + \sin \theta \tan^2 \theta}{\sec^2 \theta} \,d\theta \)

Evaluate the integral. \( \displaystyle \int^1_0 \cosh t \,dt \)

Evaluate the integral by interpreting it in terms of areas. \( \displaystyle \int^5_{-5} \bigl( x - \sqrt{25 - x^2} \bigr) \, dx \)

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

Evaluate the integral. \( \displaystyle \int^{8}_{1} \frac{2 + t}{\sqrt[3]{t^2}} \,dt \)

Evaluate the integral. \( \displaystyle \int^{\sqrt{3}}_{1/\sqrt{3}} \frac{8}{1 + x^2} \,dx \)

Evaluate the integral by interpreting it in terms of areas. \( \displaystyle \int^3_{-4} \biggl| \frac{1}{2}x \biggr| \, dx \)