91Ó°ÊÓ

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

(a) Use integration by parts to verify the validity of the reduction formula $$ \int(\ln x)^{n} d x=x(\ln x)^{n}-n \int(\ln x)^{n-1} d x $$ (b) Apply the reduction formula in (a) repeatedly to compute $$ \int(\ln x)^{3} d x $$

(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.

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

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. $$ \int g^{\prime}(x)[g(x)]^{n} d x $$

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

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

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

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. $$ \int g^{\prime}(x) \cos [g(x)] d x $$

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