Problem 82
For the following exercises, integrate using whatever method you choose.\(\int x^{3} \sqrt{x^{2}+2} d x\)
Problem 83
For the following exercises, integrate using whatever method you choose.\(\int \frac{3 x^{2}+1}{x^{4}-2 x^{3}-x^{2}+2 x} d x\)
Problem 84
For the following exercises, integrate using whatever method you choose.\(\int \frac{1}{x^{4}+4} d x\)
Problem 85
For the following exercises, integrate using whatever method you choose.\(\int \frac{\sqrt{3+16 x^{4}}}{x^{4}} d x\)
Problem 86
For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson's rule using four subintervals, rounding to three decimals.[T] \(\int_{1}^{2} \sqrt{x^{5}+2} d x\)
Problem 87
For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson's rule using four subintervals, rounding to three decimals.[T] \(\int_{0}^{\sqrt{\pi}} e^{-\sin \left(x^{2}\right)} d x\)
Problem 88
For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson's rule using four subintervals, rounding to three decimals.[I] \(\int_{1}^{4} \frac{\ln (1 / x)}{x} d x\)
Problem 89
For the following exercises, evaluate the integrals, if possible.\(\int_{1}^{\infty} \frac{1}{x^{n}} d x\), for what values of \(n\) does this integral converge or diverge?
Problem 90
For the following exercises, evaluate the integrals, if possible.\(\int_{1}^{\infty} \frac{e^{-x}}{x} d x\)
Problem 91
For the following exercises, consider the gamma function given by \(\Gamma(a)=\int_{0}^{\infty} e^{-y} y^{a-1} d y\)Show that \(\Gamma(a)=(a-1) \Gamma(a-1)\).
