91Ó°ÊÓ

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

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. $$\int \sec 4 x \, d x$$

(a).Make the indicated \(u\) -substitution, and then use the End paper 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). $$\int \frac{\cos 4 x}{9+\sin ^{2} 4 x} d x, u=\sin 4 x$$

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

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

Evaluate the integrals that converge. $$\int_{0}^{1} \frac{d x}{(x-1)^{2 / 3}}$$

Evaluate the integral. $$\int_{0}^{1} x e^{-5 x} d x$$

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

Determine whether the statement is true or false. Explain your answer. The trigonometric substitution \(x=a \sin \theta\) is made with the restriction \(0 \leq \theta \leq \pi.\)