91Ó°ÊÓ

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

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

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

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

(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. $$\int \frac{\sqrt{x^{2}-9}}{x} d x$$

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

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

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