91Ó°ÊÓ

In Problems \(37-42, a, b\), and \(c\) are constants and \(g(x)\) is a continuous function whose derivative \(g^{\prime}(x)\) is also continuous. Use substitution to evaluate each indefinite integral. $$ \int \frac{g^{\prime}(x)}{[g(x)]^{2}+1} d x $$

Use partial fraction decompositions to evaluate each integral. $$ \int \frac{x^{2}-2 x-2}{x^{2}(x+2)} \cdot d x $$

In Problems \(39-48\), first make an appropriate substitution and then use integration by parts to evaluate each integral. $$ \int x^{3} e^{-x^{2} / 2} d x $$

Find a comparison function for each integrand and determine whether the integral is convergent. $$ \int_{0}^{\infty} e^{-x^{2} / 2} d x $$

In Problems \(39-48\), first make an appropriate substitution and then use integration by parts to evaluate each integral. $$ \int x^{5} e^{x^{2}} d x $$

Find a comparison function for each integrand and determine whether the integral is convergent. $$ \int_{1}^{\infty} \frac{1}{\left(1+x^{4}\right)^{1 / 3}} d x $$

In Problems \(37-42, a, b\), and \(c\) are constants and \(g(x)\) is a continuous function whose derivative \(g^{\prime}(x)\) is also continuous. Use substitution to evaluate each indefinite integral. $$ \int g^{\prime}(x) e^{-g(x)} d x $$

Use partial fraction decompositions to evaluate each integral. $$ \int \frac{4 x^{2}-x-1}{(x+1)^{2}(x-3)} d x $$

In Problems \(39-48\), first make an appropriate substitution and then use integration by parts to evaluate each integral. $$ \int x^{3} \sin \left(x^{2}\right) d x $$

Find a comparison function for each integrand and determine whether the integral is convergent. $$ \int_{1}^{\infty} \frac{1}{\sqrt{1+x}} d x $$