91Ó°ÊÓ

Evaluate the integral. $$ \int x^{3} e^{x^{2}} d x $$

Evaluate the integral. $$ \int_{1}^{3} \frac{d x}{x^{4} \sqrt{x^{2}+3}} $$

Evaluate the integral. $$\int e^{-x} \tan \left(e^{-x}\right) d x$$

Approximate the integral using Simpson's rule \(S_{10}\) and compare your answer to that produced by a calculating utility with a numerical integration capability. Express your answers to at least four decimal places. $$ \int_{-1}^{1} e^{-x^{2}} d x $$

Evaluate the integrals by making appropriate \(u\) -substitutions and applying the formulas reviewed in this section. $$ \int \frac{e^{x}}{\sqrt{1-e^{2 x}}} d x $$

(a) Make the indicated \(u\) -substitution, and then use the Endpaper Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$\int \frac{e^{4 x}}{\left(4-3 e^{2 x}\right)^{2}} d x, u=e^{2 x}$$

Evaluate the integral. $$ \int \frac{2 x^{2}-10 x+4}{(x+1)(x-3)^{2}} d x $$

(a) Make the indicated \(u\) -substitution, and then use the Endpaper Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$\int \frac{\sin 2 x}{(\cos 2 x)(3-\cos 2 x)} d x, u=\cos 2 x$$

Evaluate the integral. $$ \int \frac{2 x^{2}-2 x-1}{x^{3}-x^{2}} d x $$

Approximate the integral using Simpson's rule \(S_{10}\) and compare your answer to that produced by a calculating utility with a numerical integration capability. Express your answers to at least four decimal places. $$ \int_{0}^{3} \frac{x}{\sqrt{2 x^{3}+1}} d x $$