Review Exercises

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

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

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

${\displaystyle \int \sqrt{x}\text{ln}(x)dx}$ using integration by parts

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

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

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

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

[T] ${\displaystyle {\int }_0^{\sqrt{\pi }}{e}^{\text{−}\text{sin}(x^2)}}dx$

${\displaystyle {\int }_1^{\infty }\frac{1}{x^n}}dx,$ for what values of $n$ does this integral converge or diverge?

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

Extend to show that $\text{Γ}(a)=(a-1)\text{!},$ 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 $v(t)=a{\text{exp}}^{bx}\text{sin}(cx)+d.$ (Hint: Consider the time units.)