91Ó°ÊÓ

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. $$ \int_{0}^{3} \frac{1}{3 x+1} d x $$

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

Evaluate the integrals by making appropriate \(u\) -substitutions and applying the formulas reviewed in this section. $$ \int \frac{1}{9+4 x^{2}} d x $$

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

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

Write out the form of the partial fraction decomposition. (Do not find the numerical values of the coefficients.) \(\frac{4 x^{3}-x}{\left(x^{2}+5\right)^{2}}\)

(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). $$ \int \frac{1}{x \sqrt{4-3 x}} d x $$

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

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