91Ó°ÊÓ

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

Evaluate the integrals that converge. $$\int_{0}^{+\infty} \frac{1}{x^{2}} d x$$

Evaluate the integral. $$\int \tan ^{2} x \sec ^{2} x d x$$

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. $$\int_{0}^{1} \cos \left(x^{2}\right) d x$$

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

(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). $$\int \frac{1}{\sqrt{4 x^{2}-9}} d x, u=2 x$$

Evaluate the integral. $$\int_{1}^{e} x^{2} \ln x d x$$

Evaluate the integral. $$\int \frac{2 x^{2}-1}{(4 x-1)\left(x^{2}+1\right)} d x$$

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

(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). $$\int x \sqrt{2 x^{4}+3} d x, u=\sqrt{2} x^{2}$$