91Ó°ÊÓ

The integral \(\int_{0}^{\infty} \frac{1}{\sqrt{x}(1+x)} d x\) is improper for two reasons: The interval \([0, \infty)\) is infinite and the integressing it as a sum of improper integrals of Type 2 and Type 1 as follows: $$\int_{0}^{\infty} \frac{1}{\sqrt{x}(1+x)} d x=\int_{0}^{1} \frac{1}{\sqrt{x}(1+x)} d x+\int_{1}^{\infty} \frac{1}{\sqrt{x}(1+x)} d x$$

Evaluate the integral. \(\int \frac{d x}{\sqrt{x^{2}+16}}\)

If \(a \neq 0\) and \(n\) is a positive integer, find the partial fraction decomposition of $$f(x)=\frac{1}{x^{n}(x-a)}$$ \([\)Hint\(:\) First find the coefficient of 1\(/(x-a) .\) Then subtract the resulting term and simplify what is left.]

Evaluate the integral. \(\int \frac{t^{5}}{\sqrt{t^{2}+2}} d t\)

Find the values of \(p\) for which the integral converges and evaluate the integral for those values of \(p .\) $$\int_{e}^{\infty} \frac{1}{x(\ln x)^{p}} d x$$

Find the values of \(p\) for which the integral converges and evaluate the integral for those values of \(p .\) $$\int_{0}^{1} \frac{1}{x^{p}} d x$$

Evaluate the integral. \(\int \sqrt{1-4 x^{2}} d x\)

(a) Evaluate the integral \(\int_{0}^{\infty} x^{n} e^{-x} d x\) for \(n=0,1,2,\) and \(3 .\) (b) Guess the value of \(\int_{0}^{\infty} x^{n} e^{-x} d x\) when \(n\) is an arbitrary positive integer. (c) Prove your guess using mathematical induction.

Evaluate the integral. \(\int \frac{d u}{u \sqrt{5-u^{2}}}\)

Evaluate the integral. \(\int \frac{\sqrt{x^{2}-9}}{x^{3}} d x\)