91Ó°ÊÓ

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 the indefinite integrals. $$ \int g^{\prime}(x) \sin [g(x)] d x $$

$$ \text { In Problems } , \text { evaluate each definite integral. } $$ $$ \int_{1}^{2} \frac{x^{2}+1}{x} d x $$

Find a comparison function for each integrand and determine whether the integral is convergent. $$ \int_{1}^{\infty} \frac{1}{\sqrt{1+x^{6}}} d x $$

First make an appropriate substitution and then use integration by parts to evaluate the indefinite integrals. $$ \int \sin \sqrt{x} d x $$

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 the indefinite integrals. $$ \int g^{\prime}(x) e^{-g(x)} d x $$

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

First make an appropriate substitution and then use integration by parts to evaluate the indefinite integrals. $$ \int x^{3} e^{-x^{2} / 2} d x $$

$$ \text { In Problems } , \text { evaluate each definite integral. } $$ $$ \int_{2}^{3} \frac{1}{1-x} d x $$

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 the indefinite integrals. $$ \int \frac{g^{\prime}(x)}{[g(x)]^{2}+1} d x $$

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