91Ó°ÊÓ

Evaluate the integrals by making a substitution (possibly trigonometric) and then applying a reduction formula. \(\int \frac{\csc ^{3} \sqrt{\theta}}{\sqrt{\theta}} d \theta\)

Evaluate each integral in Exercises \(71-82\) by using any technique you think is appropriate. $$ \int \frac{d x}{x \sqrt{3+x^{2}}} $$

a. Evaluate \(\int \cos ^{3} \theta d \theta\) . (Hint: \(\cos ^{2} \theta=1-\sin ^{2} \theta . )\) b. Evaluate \(\int \cos ^{5} \theta d \theta\) c. Without actually evaluating the integral, explain how you would evaluate \(\int \cos ^{9} \theta d \theta\)

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

Evaluate the integrals by making a substitution (possibly trigonometric) and then applying a reduction formula. \(\int_{0}^{\sqrt{3} / 2} \frac{d y}{\left(1-y^{2}\right)^{5 / 2}}\)

a. Evaluate \(\int \sin ^{3} \theta d \theta\) . (Hint: \(\sin ^{2} \theta=1-\cos ^{2} \theta . )\) b. Evaluate \(\int \sin ^{5} \theta d \theta\) c. Evaluate \(\int \sin ^{7} \theta d \theta\) d. Without actually evaluating the integral, explain how you would evaluate \(\int \sin ^{13} \theta d \theta\)

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

a. Express \(\int \tan ^{3} \theta d \theta\) in terms of \(\int \tan \theta d \theta\) . Then evaluate \(\int \tan ^{3} \theta d \theta \cdot\left(\text {Hint} : \tan ^{2} \theta=\sec ^{2} \theta-1 .\right)\) b. Express \(\int \tan ^{5} \theta d \theta\) in terms of \(\int \tan ^{3} \theta d \theta\) c. Express \(\int \tan ^{7} \theta d \theta\) in terms of \(\int \tan ^{5} \theta d \theta\) d. Express \(\int \tan ^{2 k+1} \theta d \theta,\) where \(k\) is a positive integer, in terms of \(\int \tan ^{2 k-1} \theta d \theta\)

a. Express \(\int \cot ^{3} \theta d \theta\) in terms of \(\int \cot \theta d \theta .\) Then evaluate \(\int \cot ^{3} \theta d \theta .\left(\text {Hint} : \cot ^{2} \theta=\csc ^{2} \theta-1 .\right)\) b. Express \(\int \cot ^{5} \theta d \theta\) in terms of \(\int \cot ^{3} \theta d \theta\) c. Express \(\int \cot ^{7} \theta d \theta\) in terms of \(\int \cot ^{5} \theta d \theta\) d. Express \(\int \cot ^{2 k+1} \theta d \theta,\) where \(k\) is a positive integer, in terms of \(\int \cot ^{2 k-1} \theta d \theta\)

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