91Ó°ÊÓ

Evaluate the integrals by making appropriate \(u\) -substitutions and applying the formulas reviewed in this section.$$ \int \frac{\sinh \left(x^{-1 / 2}\right)}{x^{3 / 2}} d x $$

Evaluate the integral. $$ \int_{0}^{3} \frac{x^{3}}{\left(3+x^{2}\right)^{5 / 2}} d x $$

Approximate the integral using Simpson's rule \(S_{10}\) and compare your answer to that produced by a calculating utility with a numerical integration capability. Express your answers to at least four decimal places. $$ \int_{0}^{3} \frac{x}{\sqrt{2 x^{3}+1}} d x $$

Evaluate the integral. $$ \int \frac{x e^{x}}{(x+1)^{2}} d x $$

(a) Make the indicated \(u\) -substitution, and then 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). $$\int \frac{1}{\sqrt{x}(9 x+4)} d x, u=3 \sqrt{x}$$

Determine whether the statement is true or false. Explain your answer. An integrand involving a radical of the form \(\sqrt{a^{2}-x^{2}}\) suggests the substitution \(x=a \sin \theta .\)

Evaluate the integral. $$ \int \frac{x^{2}}{(x+1)^{3}} d x $$

Evaluate the integral. $$ \int_{0}^{2} x e^{2 x} d x $$

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

Evaluate the integral. $$\int \sec 4 x d x$$