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. $$\int_{0}^{2} \sin x d x$$

(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. $$\int \sin ^{3} a \theta d \theta$$

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

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

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

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