91Ó°ÊÓ

Determine whether the statement is true or false. Explain your answer. Applying the LIATE strategy to evaluate \(\int x^{3} \ln x d x,\) we should choose \(u=x^{3}\) and \(d v=\ln x d x\)

Evaluate the integral by making a substitution that converts the integrand to a rational function. $$ \int \frac{e^{t}}{e^{2 t}-4} d t $$

Make the \(u\) -substitution and evaluate the resulting definite integral. $$ \int_{0}^{+\infty} \frac{e^{-x}}{\sqrt{1-e^{-2 x}}} d x ; u=e^{-x} $$

Determine whether the statement is true or false. Explain your answer. To evaluate \(\int \ln e^{x} d x\) using integration by parts, choose \(d v=e^{x} d x .\)

Evaluate the integral. $$ \int \frac{d x}{\sqrt{x^{2}-6 x+10}} $$

(a) Make an appropriate u-substitution, and then use the Endpaper Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral (no substitution), and then confirm that the result is equivalent to that in part (a). $$\int e^{x} \sqrt{3-4 e^{2 x}} d x$$

Express the improper integral as a limit, and then evaluate that limit with a CAS. Confirm the answer by evaluating the integral directly with the CAS. $$ \int_{0}^{+\infty} e^{-x} \cos x d x $$

Evaluate the integral by making a substitution that converts the integrand to a rational function. $$ \int \frac{e^{3 x}}{e^{2 x}+4} d x $$

Evaluate the integral. $$\int \tan ^{3} 4 x d x$$

Evaluate the integral by making a substitution that converts the integrand to a rational function. $$ \int \frac{5+2 \ln x}{x(1+\ln x)^{2}} d x $$