91Ó°ÊÓ

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

(a) Use the Endpaper 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. $$ \int x e^{-2 x} d x $$

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

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) \(\int_{-\infty}^{+\infty} \frac{d x}{\sqrt[3]{x-1}}\) (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. $$ \int_{0}^{3} \sqrt{x+1} d x $$

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

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

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) \(\int_{0}^{1} e^{-p x} d x\)

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