91Ó°ÊÓ

Use integration by parts to establish the reduction formula. $$\int x^{n} \cos x d x=x^{n} \sin x-n \int x^{n-1} \sin x d x$$

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. $$\int_{-\infty}^{\infty} \frac{d x}{\sqrt{x^{4}+1}}$$

Use integration by parts to establish the reduction formula. $$\int x^{n} \sin x d x=-x^{n} \cos x+n \int x^{n-1} \cos x d x$$

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. $$\int_{-\infty}^{\infty} \frac{d x}{e^{x}+e^{-x}}$$

Use any method to evaluate the integrals. $$\int \frac{\sin ^{3} x}{\cos ^{4} x} d x$$

Use a CAS to perform the integrations. Evaluate the integrals a. \(\int x \ln x \, d x \quad\) b. \, \(\int x^{2} \ln x d x \quad\) c. \(\int x^{3} \ln x d x\) d. What pattern do you see? Predict the formula for \(\int x^{4} \ln x d x\) and then see if you are correct by evaluating it with a CAS. e. What is the formula for \(\int x^{n} \ln x d x, n \geq 1 ?\) Check your answer using a CAS.

Use any method to evaluate the integrals. $$\int \frac{\tan ^{2} x}{\csc x} d x$$

Find the values of \(p\) for which each integral converges. a. \(\int_{1}^{2} \frac{d x}{x(\ln x)^{p}}\) b. \(\int_{2}^{\infty} \frac{d x}{x(\ln x)^{p}}\)

Use any method to evaluate the integrals. $$\int \frac{\cot x}{\cos ^{2} x} d x$$

\(\int_{-\infty}^{\infty} f(x) d x\) may not equal \(\lim _{b \rightarrow \infty} \int_{-b}^{b} f(x) d x \quad\) Show that $$\int_{0}^{\infty} \frac{2 x d x}{x^{2}+1}$$ diverges and hence that $$\int_{-\infty}^{\infty} \frac{2 x d x}{x^{2}+1}$$ diverges. Then show that $$\lim _{b \rightarrow \infty} \int_{-b}^{b} \frac{2 x d x}{x^{2}+1}=0.$$