91Ó°ÊÓ

Evaluate the following integrals or state that they diverge. $$\int_{0}^{\ln 3} \frac{e^{y}}{\left(e^{y}-1\right)^{2 / 3}} d y$$

A differential equation and its direction field are given. Sketch a graph of the solution that results with each initial condition. $$\begin{aligned}&y^{\prime}(t)=\frac{t^{2}}{y^{2}+1},\\\&y(0)=-2 \text { and }\\\&y(-2)=0\end{aligned}$$

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

Determine whether the following statements are true and give an explanation or counterexample. a. The Trapezoid Rule is exact when used to approximate the definite integral of a linear function. b. If the number of subintervals used in the Midpoint Rule is increased by a factor of \(3,\) the error is expected to decrease by a factor of 8. c. If the number of subintervals used in the Trapezoid Rule is increased by a factor of \(4,\) the error is expected to decrease by a factor of 16.

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

Explain why or why not 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. \(\int v d u=u v-\int u d v\)

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

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

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