91影视

Evaluate the integrals. If the integral diverges, answer "diverges."\(\int_{0}^{e} \ln (x) d x\)

State the method of integration you would use to evaluate the integral \(\int x \sqrt{x^{2}+1} d x\). Why did you choose this method?

Use tables to perform the integration. $$ \int \frac{d x}{\sqrt{4 x+1}} $$

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.) \(\int \sec ^{4} x d x\)

Compute the definite integrals. Use a graphing utility to confirm your answers. $$ \int_{0}^{\pi / 2} x^{2} \sin x d x \text { (Express the answer in exact form.) } $$

Find an upper bound for the error in estimating \(\int_{0}^{3}(5 x+4) d x\) using the trapezoidal rule with six steps.

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. \(\int_{0}^{1} \frac{e^{x}}{36-e^{2 x}} d x\) (Give the exact answer and the decimal equivalent. Round to five decimal places.)

State the method of integration you would use to evaluate the integral \(\int x^{2} \sqrt{x^{2}-1} d x .\) Why did you choose this method?

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. \(\int \frac{e^{x} d x}{e^{2 x}-e^{x}} d x\)

Compute the definite integrals. Use a graphing utility to confirm your answers. $$ \int_{0}^{1} x 5^{x} d x \text { (Express the answer using five significant digits.) } $$