91Ó°ÊÓ

$$ \text { In Problems } , \text { evaluate each integral. } $$ $$ \int \frac{x^{3}+1}{x^{2}+3} d x $$

$$ \text { In Problems } , \text { evaluate each integral. } $$ $$ \int \frac{x^{2}+4}{x^{2}-4} d x $$

Use substitution to evaluate the indefinite integrals. $$ \int x^{3} \sqrt{5+x^{2}} d x $$

Use integration by parts to show that $$ \int x^{n} e^{x} d x=x^{n} e^{x}-n \int x^{n-1} e^{x} d x $$ Such formulas are called reduction formulas, since they reduce the exponent of \(x\) by 1 each time they are applied. (b) Apply the reduction formula in (a) repeatedly to compute $$ \int x^{3} e^{x} d x $$

(a) Show that $$ 0 \leq e^{-x^{2}} \leq e^{-x} $$ for \(x \geq 1\). (b) Use your result in (a) to show that $$ \int_{1}^{\infty} e^{-x^{2}} d x $$ is convergent.

$$ \text { In Problems } , \text { evaluate each integral. } $$ $$ \int \frac{x^{4}+3}{x^{2}-4 x+3} d x $$

Use substitution to evaluate the indefinite integrals. $$ \int \sqrt{1+\ln x} \frac{\ln x}{x} d x $$

(a) Show that $$ 0 \leq \frac{1}{\sqrt{1+x^{4}}} \leq \frac{1}{x^{2}} $$ for \(x>0\). (b) Use your result in (a) to show that $$ \int_{1}^{\infty} \frac{1}{\sqrt{1+x^{4}}} d x $$ is convergent.

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 the indefinite integrals. $$ \int \frac{2 a x+b}{a x^{2}+b x+c} d x $$

$$ \text { In Problems } 37-44, \text { evaluate each definite integral. } $$ $$ \int_{3}^{5} \frac{x-1}{x} d x $$