91Ó°ÊÓ

Find the value of the constant \(C\) for which the integral $$ \int_{0}^{\infty}\left(\frac{1}{\sqrt{x^{2}+4}}-\frac{C}{x+2}\right) d x $$ converges. Evaluate the integral for this value of \(C .\)

Evaluate the integral. $$ \int x \sin ^{2} x \cos x d x $$

Find the value of the constant \(C\) for which the integral $$ \int_{0}^{\infty}\left(\frac{x}{x^{2}+1}-\frac{C}{3 x+1}\right) d x $$ converges. Evaluate the integral for this value of \(C .\)

Evaluate the integral. $$ \int \frac{\sec x \cos 2 x}{\sin x+\sec x} d x $$

Evaluate the integral. $$ \int \sqrt{1-\sin x} d x $$

Suppose \(f\) is continuous on \([0, \infty)\) and \(\lim _{x \rightarrow \infty} f(x)=1 .\) Is it possible that \(\int_{0}^{\infty} f(x) d x\) is convergent?

Evaluate the integral. $$ \int \frac{\sin x \cos x}{\sin ^{4} x+\cos ^{4} x} d x $$

Show that if \(a>-1\) and \(b>a+1\), then the following integral is convergent. $$ \int_{0}^{\infty} \frac{x^{a}}{1+x^{b}} d x $$

The functions \(y=e^{x^{2}}\) and \(y=x^{2} e^{x^{2}}\) don't have elementary antiderivatives, but \(y=\left(2 x^{2}+1\right) e^{x^{2}}\) does. Evaluate \(\int\left(2 x^{2}+1\right) e^{x^{2}} d x\)

We know that \(F(x)=\int_{0}^{x} e^{\varepsilon^{t}} d t\) is a continuous function by FTC1, though it is not an elementary function. The functions $$ \int \frac{e^{x}}{x} d x \quad \text { and } \quad \int \frac{1}{\ln x} d x $$ are not elementary either, but they can be expressed in terms of F. Evaluate the following integrals in terms of F. $$ \text { (a) } \int_{1}^{2} \frac{e^{x}}{x} d x \quad \text { (b) } \int_{2}^{3} \frac{1}{\ln x} d x $$