91Ó°ÊÓ

Evaluate the integrals in Exercises \(51-56\) $$ \int_{-\pi}^{\pi} \sin 3 x \sin 3 x d x $$

Use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{1}^{\infty} \frac{\sqrt{x+1}}{x^{2}} d x$$

Area Find the area of the region in the first quadrant that is enclosed by the coordinate axes and the curve \(y=\sqrt{9-x^{2}} / 3.\)

Solve the initial value problems in Exercises \(51-54\) for \(x\) as a function of \(t .\) $$\left(t^{2}+2 t\right) \frac{d x}{d t}=2 x+2 \quad(t, x>0), \quad x(1)=1$$

Evaluate the integrals by making a substitution (possibly trigonometric) and then applying a reduction formula. \(\int_{0}^{1} 2 \sqrt{x^{2}+1} d x\)

\begin{equation} \begin{array}{l}{\text { Finding area Find the area of the region enclosed by the curve }} \\ {y=x \cos x \text { and the } x \text { -axis ( see the accompanying figure) for }} \\ {\text { a. } \pi / 2 \leq x \leq 3 \pi / 2} \\\ {\text { b. } 3 \pi / 2 \leq x \leq 5 \pi / 2} \\ {\text { c. } 5 \pi / 2 \leq x \leq 7 \pi / 2} \\ {\text { d. What pattern do you see? What is the area between the curve }} \\ {\text { and the } x \text { -axis for }}\end{array} \end{equation} $$ \left(\frac{2 n-1}{2}\right) \pi \leq x \leq\left(\frac{2 n+1}{2}\right) \pi $$ \(n\) an arbitrary positive integer? Give reasons for your answer.

Evaluate the integrals in Exercises \(51-56\) $$ \int_{0}^{\pi / 2} \sin x \cos x d x $$

Solve the initial value problems in Exercises \(51-54\) for \(x\) as a function of \(t .\) $$(t+1) \frac{d x}{d t}=x^{2}+1 \quad(t>-1), \quad x(0)=0$$

Use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{2}^{\infty} \frac{x d x}{\sqrt{x^{4}-1}}$$

Area Find the area enclosed by the ellipse $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1.$$