91Ó°ÊÓ

Evaluate the integral. $$ \int \frac{x^{3}+3 x^{2}+x+9}{\left(x^{2}+1\right)\left(x^{2}+3\right)} d x $$

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

Evaluate the integrals that converge. $$ \int_{0}^{1} \frac{d x}{\sqrt{x}(x+1)} $$

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

The integral $$ \int \frac{x^{2}}{x^{2}+4} d x $$ can be evaluated either by a trigonometric substitution or by algebraically rewriting the numerator of the integrand as \(\left(x^{2}+4\right)-4 .\) Do it both ways and show that the results are equivalent.

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

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

(a) Derive the identity $$ \frac{\operatorname{sech}^{2} x}{1+\tanh ^{2} x}=\operatorname{sech} 2 x $$ (b) Use the result in part (a) to evaluate \(\int \operatorname{sech} x d x\). (c) Derive the identity $$ \operatorname{sech} x=\frac{2 e^{x}}{e^{2 x}+1} $$ (d) Use the result in part (c) to evaluate \(\int \operatorname{sech} x d x\) (e) Explain why your answers to parts \((b)\) and \((d)\) 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). $$\int \frac{1}{x^{2} \sqrt{3-4 x^{2}}} d x, u=2 x$$

Find the arc length of the curve \(y=\ln x\) from \(x=1\) to \(x=2 .\)