91Ó°ÊÓ

Use reduction formulas to evaluate the integrals. \(\int 2 \sec ^{3} \pi x d x\)

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

Evaluate each integral in Exercises \(63-70\) by eliminating the square root. $$ \int_{\pi / 2}^{\pi} \sqrt{1+\cos 2 t} d t $$

Use reduction formulas to evaluate the integrals. \(\int \frac{1}{2} \csc ^{3} \frac{x}{2} d x\)

\(\int_{-\infty}^{\infty} f(x) d x\) may not equal \(\lim _{b \rightarrow \infty} \int_{-b}^{b} f(x) d x\) 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 $$\lim _{b \rightarrow \infty} \int_{-b}^{b} \frac{2 x d x}{x^{2}+1}=0$$

Evaluate each integral in Exercises \(63-70\) by eliminating the square root. $$ \int_{-\pi}^{0} \sqrt{1+\cos t} d t $$

Exercises \(67-70\) are about the infinite region in the first quadrant between the curve \(y=e^{-x}\) and the \(x\) -axis. Find the area of the region.

Use reduction formulas to evaluate the integrals. \(\int 3 \sec ^{4} 3 x d x\)

Evaluate each integral in Exercises \(63-70\) by eliminating the square root. $$ \int_{-\pi}^{0} \sqrt{1-\cos ^{2} \theta} d \theta $$

Evaluate each integral in Exercises \(63-70\) by eliminating the square root. $$ \int_{\pi / 2}^{\pi} \sqrt{1-\sin ^{2} \theta} d \theta $$