91Ó°ÊÓ

Perform the indicated integrations. $$ \int \tan ^{1 / 2} x \sec ^{4} x d x $$

Perform the indicated integrations. $$ \int_{0}^{\pi / 6} 2^{\cos x} \sin x d x $$

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{x^{2}+19 x+10}{2 x^{4}+5 x^{3}} d x\)

The region bounded by \(y=1 /\left(x^{2}+2 x+5\right), y=0\), \(x=0\), and \(x=1\), is revolved about the \(x\) -axis. Find the volume of the resulting solid.

Perform the indicated integrations. $$ \int \frac{\sin x-\cos x}{\sin x} d x $$

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{2 x^{2}+x-8}{x^{3}+4 x} d x\)

Use the table of integrals on the inside back cover, perhaps combined with a substitution, to evaluate the given integrals. $$ \int \frac{\cos t \sin t}{\sqrt{2 \cos t+1}} d t $$

Perform the indicated integrations. $$ \int \tan ^{3} x \sec ^{2} x d x $$

For the differential equation \(\frac{d y}{d x}-\frac{y}{x}=x^{2}, x>0\), the integrating factor is \(e^{\int(-1 / x) d x} .\) The general antiderivative \(\int\left(-\frac{1}{x}\right) d x\) is equal to \(-\ln x+C .\) (a) Multiply both sides of the differential equation by \(\exp \left(\int\left(-\frac{1}{x}\right) d x\right)=\exp (-\ln x+C), \quad\) and show that \(\exp (-\ln x+C)\) is an integrating factor for every value of \(C .\) (b) Solve the resulting equation for \(y\), and show that the solution agrees with the solution obtained when we assumed that \(C=0\) in the integrating factor.

Multiply both sides of the equation \(\frac{d y}{d x}+P(x) y=Q(x)\) by the factor \(e^{\int P(x) d x+C}\). (a) Show that \(e^{\int P(x) d x+C}\) is an integrating factor for every value of \(C\). (b) Solve the resulting equation for \(y\), and show that it agrees with the general solution given before Example 1 .