### Review Exercises

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.

$\int {e}^{x}\text{sin}(x)dx$ cannot be integrated by parts.

In numerical integration, increasing the number of points decreases the error.

For the following exercises, evaluate the integral using the specified method.

$\int {x}^{2}\text{sin}(4x)dx$ using integration by parts

$\int \sqrt{x}\phantom{\rule{0.1em}{0ex}}\text{ln}(x)dx$ using integration by parts

$\int \frac{{x}^{5}}{{\left(4{x}^{2}+4\right)}^{5\text{/}2}}dx$ using trigonometric substitution

$\int \frac{\sqrt{4-{\text{sin}}^{2}(x)}}{{\text{sin}}^{2}(x)}\text{cos}(x)dx$ using a table of integrals or a CAS

For the following exercises, integrate using whatever method you choose.

$\int {\text{sin}}^{2}(x){\text{cos}}^{2}(x)dx$

$\int \frac{3{x}^{2}+1}{{x}^{4}-2{x}^{3}-{x}^{2}+2x}dx$

$\int \frac{\sqrt{3+16{x}^{4}}}{{x}^{3}}dx$

For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson’s rule using four subintervals, rounding to three decimals.

**[T]** ${\int}_{0}^{\sqrt{\pi}}{e}^{\text{\u2212}\text{sin}({x}^{2})}}dx$

For the following exercises, evaluate the integrals, if possible.

${\int}_{1}^{\infty}\frac{1}{{x}^{n}}}dx,$ for what values of $n$ does this integral converge or diverge?

For the following exercises, consider the gamma function given by $\text{\Gamma}(a)={\displaystyle {\int}_{0}^{\infty}{e}^{\text{\u2212}y}{y}^{a-1}dy}.$

Show that $\text{\Gamma}(a)=(a-1)\text{\Gamma}(a-1).$

Extend to show that $\text{\Gamma}(a)=(a-1)\text{!},$ assuming $a$ is a positive integer.

The fastest car in the world, the Bugati Veyron, can reach a top speed of 408 km/h. The graph represents its velocity.

**[T]** Use the graph to estimate the velocity every 20 sec and fit to a graph of the form $v(t)=a\phantom{\rule{0.1em}{0ex}}{\text{exp}}^{bx}\text{sin}(cx)+d.$ (*Hint:* Consider the time units.)

**[T]** Using your function from the previous problem, find exactly how far the Bugati Veyron traveled in the 1 min 40 sec included in the graph.