91Ó°ÊÓ

Use a table of integrals to solve the following problems. The region bounded by the graph of \(y=1 /(x+10)\) and the \(x\) -axis on the interval [0,3] is revolved about the \(x\) -axis. What is the volume of the solid that is formed?

Evaluate the following integrals. $$\int \frac{d x}{x^{3} \sqrt{x^{2}-100}}, x>10$$

Evaluate the following integrals or state that they diverge. $$\int_{0}^{1} x^{2} \ln (1 / x) d x$$

Compare the errors in the Midpoint and Trapezoid Rules with \(n=4.8 .\) If. and 32 subintervals when they are applied to the following integrals (with their exact values given). $$\int_{0}^{1}\left(8 x^{7}-7 x^{8}\right) d x=\frac{2}{9}$$

Evaluate the following integrals. $$\int \frac{x^{2}+x+2}{(x+1)\left(x^{2}+1\right)} d x$$

Use the approaches discussed in this section to evaluate the following integrals. $$\int_{-1}^{0} \frac{x}{x^{2}+2 x+2} d x$$

Evaluate the following integrals. $$\int \frac{x^{3}}{\left(81-x^{2}\right)^{2}} d x$$

Determine whether the following statements are true and give an explanation or counterexample. a. \(\int u v^{\prime} d x=\left(\int u d x\right)\left(\int v^{\prime} d x\right)\) b. \(\int u v^{\prime} d x=u v-\int v u^{\prime} d x\) c. \(\int v d u=u v-\int u d v\)

Evaluate the following integrals. $$\int_{\pi / 6}^{\pi / 3} \cot ^{3} \theta d \theta$$

Evaluate the following integrals or state that they diverge. $$\int_{0}^{1} \frac{x^{3}}{x^{4}-1} d x$$