91Ó°ÊÓ

Use the approaches discussed in this section to evaluate the following integrals. $$\int_{0}^{\pi / 2} \sqrt{1+\cos 2 x} d x$$

Use the window \([-2,2] \times[-2,2]\) to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. $$y^{\prime}(t)=y-3, y(0)=1$$

Evaluate the following integrals. $$\int \frac{x^{3}}{\left(x^{2}-16\right)^{3 / 2}} d x, x<-4$$

Evaluate the following integrals or state that they diverge. $$\int_{1}^{11} \frac{d x}{(x-3)^{2 / 3}}$$

Evaluate the following integrals. $$\int \frac{z+1}{z\left(z^{2}+4\right)} d z$$

Integrals of cot \(x\) and \(\csc x\) Use a change of variables to prove that \(\int \cot x d x=\ln |\sin x|+C\).

Use a table of integrals to solve the following problems. The graphs of \(f(x)=\frac{2}{x^{2}+1}\) and \(g(x)=\frac{7}{4 \sqrt{x^{2}+1}}\) are shown in the figure. Which is greater, the average value of \(f\) or that of \(g\) on the interval [-1,1]\(?\)

Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. $$\int \frac{x}{\sqrt{2 x+3}} d x$$

Use the window \([-2,2] \times[-2,2]\) to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. $$y^{\prime}(x)=\sin x, y(-2)=2$$

Evaluate the following definite $$\int_{0}^{1} \frac{d x}{\sqrt{x^{2}+16}}$$