91Ó°ÊÓ

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

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

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) \(\int_{0}^{\pi / 4} \tan x d x\)

Write out the form of the partial fraction decomposition. (Do not find the numerical values of the coefficients.) $$\frac{3 x-1}{(x-2)(x+5)}$$

(a) Use the Endpaper Integral Table to evaluate the 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). $$\int \frac{3 x}{4 x-1} d x$$

Use \(n=10\) subdivisions to approximate the integral by (a) the midpoint rule, (b) the trapezoidal rule. and (c) Simpson's rule. 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. $$\int \sqrt{4-x^{2}} d x$$

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

Evaluate the integrals by making appropriate \(u\) -substitutions and applying the formulas reviewed in this section. $$\int \sqrt{4+9 x} d x$$

Use \(n=10\) subdivisions to approximate the integral by (a) the midpoint rule, (b) the trapezoidal rule. and (c) Simpson's rule. In each case find the exact value of the integral and approximate the absolute error. Express your answers to at least four decimal places. $$\int_{1}^{4} \frac{1}{\sqrt{x}} d x$$