91Ó°ÊÓ

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

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

Determine whether the statement is true or false. Explain your answer. The area enclosed by the ellipse \(x^{2}+4 y^{2}=1\) is \(\pi / 2\)

Evaluate the integral. $$\int \tan ^{5} x \sec ^{4} x d x$$

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

The exact value of the given integral is \(\pi\) (verify). 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). Approximate the ab- solute error and express your answers to at least four decimal places. $$ \int_{0}^{2} \frac{8}{x^{2}+4} d x $$

Evaluate the integral. $$ \int_{-1}^{1} \ln (x+2) d x $$

The integral $$ \int \frac{x}{x^{2}+4} d x $$ can be evaluated either by a trigonometric substitution or by the substitution \(u=x^{2}+4 .\) Do it both ways and show that the results are equivalent.

(a) Evaluate the integral \(\int \sin x \cos x d x\) using the sub- stitution \(u=\sin x\). (b) Evaluate the integral \(\int \sin x \cos x d x\) using the iden- tity \(\sin 2 x=2 \sin x \cos x .\) (c) Explain why your answers to parts (a) and (b) are consistent.

(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). $$\int \frac{4 x^{5}}{\sqrt{2-4 x^{4}}} d x, u=2 x^{2}$$