91Ó°ÊÓ

Evaluate the following improper integrals whenever they are convergent. $$\int_{0}^{\infty} \frac{x}{x^{2}+1} d x$$

Approximate the integrals by the midpoint rule, the trapezoidal rule, and Simpson's rule with \(n=10 .\) Then, find the exact value by integration and give the error for each approximation. Express your answers to the full accuracy given by the calculator or computer. $$\int_{1}^{11} \frac{1}{x} d x$$

Evaluate the following improper integrals whenever they are convergent. $$\int_{0}^{\infty} 2 x\left(x^{2}+1\right)^{-3 / 2} d x$$

Evaluate the following improper integrals whenever they are convergent. $$\int_{1}^{\infty}(5 x+1)^{-4} d x$$

Approximate the integrals by the midpoint rule, the trapezoidal rule, and Simpson's rule with \(n=10 .\) Then, find the exact value by integration and give the error for each approximation. Express your answers to the full accuracy given by the calculator or computer. $$\int_{0}^{\pi / 2} \cos x d x$$

Evaluate the following improper integrals whenever they are convergent. $$\int_{-\infty}^{0} e^{4 x} d x$$

Evaluate $$\int \frac{x e^{x}}{(x+1)^{2}} d x$$ using integration by parts. \(\left[\text{Hint:} .f(x)=x e^{x}, g(x)=\frac{1}{(x+1)^{2}}\right]\).

Determine the following integrals using the indicated substitution. $$\int(x+5)^{-1 / 2} e^{\sqrt{x+5}} d x ; u=\sqrt{x+5}$$

Approximate the integrals by the midpoint rule, the trapezoidal rule, and Simpson's rule with \(n=10 .\) Then, find the exact value by integration and give the error for each approximation. Express your answers to the full accuracy given by the calculator or computer. $$\int_{0}^{1} 2 x e^{x^{2}} d x$$

Determine the following integrals using the indicated substitution. $$\int \frac{x^{4}}{x^{5}-7} \ln \left(x^{5}-7\right) d x ; u=\ln \left(x^{5}-7\right)$$