91Ó°ÊÓ

Evaluate the integrals $$\int_{-\pi}^{\pi} \sin 3 x \sin 3 x d x$$

Evaluate the integrals. Some integrals do not require integration by parts. $$\int \sqrt{x}\left(\sin ^{-1} \sqrt{x}\right) d x$$

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. $$\int_{0}^{\infty} \frac{d x}{\sqrt{x^{6}+1}}$$

Evaluate the integrals by making a substitution (possibly trigonometric) and then applying a reduction formula. $$\int_{1}^{2} \frac{\left(r^{2}-1\right)^{3 / 2}}{r} d r$$

Use any method to evaluate the integrals in Exercises \(55-66\) $$\int \frac{x^{3}-2 x^{2}-3 x}{x+2} d x$$

Evaluate the integrals by making a substitution (possibly trigonometric) and then applying a reduction formula. $$\int_{0}^{1 / \sqrt{3}} \frac{d t}{\left(t^{2}+1\right)^{7 / 2}}$$

Evaluate the integrals. Some integrals do not require integration by parts. $$\int \frac{\left(\sin ^{-1} x\right)^{2}}{\sqrt{1-x^{2}}} d x$$

Solve the initial value problems in Exercises \(53-56\) for \(y\) as a function of \(x\). $$\left(x^{2}+1\right)^{2} \frac{d y}{d x}=\sqrt{x^{2}+1}, \quad y(0)=1$$

Evaluate the integrals $$\int_{0}^{\pi / 2} \sin x \cos x d x$$

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. $$\int_{2}^{\infty} \frac{d x}{\sqrt{x^{2}-1}}$$