Chapter 7

In each part, determine whether the integral is improper, and if so, explain why. (a) \(\int_{1}^{5} \frac{d x}{x-3}\) (b) \(\int_{1}^{5} \frac{d x}{x+3}\) (c) \(\int_{0}^{1} \ln x d x\) (d) \(\int_{1}^{+\infty} e^{-x} d x\) (e) \(\int_{-\infty}^{+\infty} \frac{d x}{\sqrt[3]{x-1}}(\mathrm {f}) \int_{0}^{\pi / 4} \tan x d x\)

Approximate the integral using (a) the midpoint approximation \(M_{10},\) (b) the trapezoidal approximation \(T_{10},\) and (c) Simpson's rule approximation \(S_{20}\) using Formula (7). In each case, find the exact value of the integral and approximate the absolute error. Express your answers to at least four decimal places. $$\int_{0}^{3} \sqrt{x+1} d x$$

(a) Use the End paper Integral Table to evaluate the given integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$\int \frac{4 x}{3 x-1} d x$$

Evaluate the integral. $$\int \sqrt{4-x^{2}} d x$$

Write out the form of the partial fraction decomposition. (Do not find the numerical values of the coefficients.) $$\frac{3 x-1}{(x-3)(x+4)}$$

Evaluate the integral. $$\int x e^{-2 x} d x$$

Evaluate the integral. $$\int \cos ^{3} x \sin x d x$$

Evaluate the integrals by making appropriate \(u\) -substitutions and applying the formulas reviewed in this section. $$\int(4-2 x)^{3} d x$$

In each part, determine all values of \(p\) for which the integral is improper. (a) \(\int_{0}^{1} \frac{d x}{x^{p}}\) (b) \(\int_{1}^{2} \frac{d x}{x-p}\) (c) \(\int_{0}^{1} e^{-p x} d x\)

Approximate the integral using (a) the midpoint approximation \(M_{10},\) (b) the trapezoidal approximation \(T_{10},\) and (c) Simpson's rule approximation \(S_{20}\) using Formula (7). In each case, find the exact value of the integral and approximate the absolute error. Express your answers to at least four decimal places. $$\int_{4}^{9} \frac{1}{\sqrt{x}} d x$$

Evaluate the integral. $$\int \sin ^{5} 3 x \cos 3 x d x$$

(a) Use the End paper Integral Table to evaluate the given integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$\int \frac{x}{(4-5 x)^{2}} d x$$

Evaluate the integral. $$\int x e^{3 x} d x$$

Evaluate the integral. $$\int \sqrt{1-4 x^{2}} d x$$

Write out the form of the partial fraction decomposition. (Do not find the numerical values of the coefficients.) $$\frac{5}{x\left(x^{2}-4\right)}$$

Evaluate the integrals by making appropriate \(u\) -substitutions and applying the formulas reviewed in this section. $$\int 3 \sqrt{4+2 x} d x$$

Evaluate the integrals that converge. $$\int_{0}^{+\infty} e^{-2 x} d x$$

Approximate the integral using (a) the midpoint approximation \(M_{10},\) (b) the trapezoidal approximation \(T_{10},\) and (c) Simpson's rule approximation \(S_{20}\) using Formula (7). In each case, find the exact value of the integral and approximate the absolute error. Express your answers to at least four decimal places. $$\int_{0}^{\pi / 2} \cos x d x$$

Evaluate the integral. $$\int x^{2} e^{x} d x$$

Evaluate the integral. $$\int \sin ^{2} 5 \theta d \theta$$

(a) Use the End paper Integral Table to evaluate the given integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$\int \frac{1}{x(2 x+5)} d x \quad$$

Evaluate the integral. $$\int \frac{x^{2}}{\sqrt{16-x^{2}}} d x$$

Write out the form of the partial fraction decomposition. (Do not find the numerical values of the coefficients.) $$\frac{2 x-3}{x^{3}-x^{2}}$$

Evaluate the integrals by making appropriate \(u\) -substitutions and applying the formulas reviewed in this section. $$\int x \sec ^{2}\left(x^{2}\right) d x$$

Approximate the integral using (a) the midpoint approximation \(M_{10},\) (b) the trapezoidal approximation \(T_{10},\) and (c) Simpson's rule approximation \(S_{20}\) using Formula (7). In each case, find the exact value of the integral and approximate the absolute error. Express your answers to at least four decimal places. $$\int_{0}^{2} \sin x d x$$

Evaluate the integrals that converge. $$\int_{-1}^{+\infty} \frac{x}{1+x^{2}} d x$$

Evaluate the integral. $$\int x^{2} e^{-2 x} d x$$

Evaluate the integral. $$\int \cos ^{2} 3 x d x$$

Evaluate the integral. $$\int \frac{d x}{x^{2} \sqrt{9-x^{2}}}$$

Write out the form of the partial fraction decomposition. (Do not find the numerical values of the coefficients.) $$\frac{x^{2}}{(x+2)^{3}}$$

Evaluate the integrals by making appropriate \(u\) -substitutions and applying the formulas reviewed in this section. $$\int 4 x \tan \left(x^{2}\right) d x$$

Approximate the integral using (a) the midpoint approximation \(M_{10},\) (b) the trapezoidal approximation \(T_{10},\) and (c) Simpson's rule approximation \(S_{20}\) using Formula (7). In each case, find the exact value of the integral and approximate the absolute error. Express your answers to at least four decimal places. $$\int_{1}^{3} e^{-2 x} d x$$

Evaluate the integrals that converge. $$\int_{3}^{+\infty} \frac{2}{x^{2}-1} d x$$

Evaluate the integral. $$\int x \sin 3 x \, d x$$

(a) Use the End paper Integral Table to evaluate the given integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$\int x \sqrt{2 x+3} d x$$

Evaluate the integral. $$\int \sin ^{3} a \theta d \theta$$

Evaluate the integral. $$\int \frac{d x}{\left(4+x^{2}\right)^{2}}$$

Write out the form of the partial fraction decomposition. (Do not find the numerical values of the coefficients.) $$\frac{1-x^{2}}{x^{3}\left(x^{2}+2\right)}$$

Evaluate the integrals by making appropriate \(u\) -substitutions and applying the formulas reviewed in this section. $$\int \frac{\sin 3 x}{2+\cos 3 x} d x$$

Evaluate the integrals that converge. $$\int_{0}^{+\infty} x e^{-x^{2}} d x$$

Approximate the integral using (a) the midpoint approximation \(M_{10},\) (b) the trapezoidal approximation \(T_{10},\) and (c) Simpson's rule approximation \(S_{20}\) using Formula (7). In each case, find the exact value of the integral and approximate the absolute error. Express your answers to at least four decimal places. $$\int_{0}^{3} \frac{1}{3 x+1} d x$$

Evaluate the integral. $$\int x \cos 2 x \, d x$$

(a) Use the End paper Integral Table to evaluate the given integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$\int \frac{x}{\sqrt{2-x}} d x$$

Evaluate the integral. $$\int \cos ^{3} a t d t$$

Evaluate the integral. $$\int \frac{x^{2}}{\sqrt{5+x^{2}}} d x$$

Write out the form of the partial fraction decomposition. (Do not find the numerical values of the coefficients.) $$\frac{3 x}{(x-1)\left(x^{2}+6\right)}$$

Evaluate the integrals that converge. $$\int_{e}^{+\infty} \frac{1}{x \ln ^{3} x} d x$$

Evaluate the integral. $$\int x^{2} \cos x d x$$

(a) Use the End paper Integral Table to evaluate the given integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$\int \frac{1}{x \sqrt{4-3 x}} d x$$

Evaluate the integral. $$\int \sin a x \cos a x \, d x$$