Problems

01

Trig sub

Compute the definite integral:

$${\int }_0^{1/2}\frac{x^2}{\sqrt{1-x^2}}dx$$

02

Trig sub

Compute the integral:

$$\int \frac{dx}{\sqrt{x^2+4}}$$

03

Trig sub

Compute the integral:

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

04

Trig sub

Compute the integral:

$$\int \frac{dx}{\sqrt{x^2+4x+13}}$$

Hint: complete the square and then substitute.

05

Trig sub

Compute the integral:

$$\int \frac{x^2}{{(x^2+1)}^{3/2}}dx$$

06

Double sub: $u$-sub then trig sub

Compute the definite integral:

$${\int }_0^{\pi /2}\frac{\cos x}{\sqrt{1+{\sin }^{2}x}}dx$$

07

Trig sub for electric charge

A charged wire lies on the $x$-axis running from $x_1$ to $x_2$. The electric field at the point $p=(0,h)$ is given by:

$$E={\int }_{x_1}^{x_2}\frac{k\lambda h}{{(x^2+h^2)}^{3/2}}dx$$

Find the numerical value of $E$ assuming $\lambda =6.0\times {10}^{-4}$ and $k=8.99\times {10}^9$ and $h=3$ and $(x_1,x_2)=(-\mathrm{15,15})$.