W02 Regular

Due date: Sunday 1/25, 11:59pm

Trig power products

01 $★$

04

All odd power product

Compute the integral:

$$\int {\cos }^{7}xdx$$

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Solution

Solutions---0040-04

02 $★$

05

Tangent and secant mixed parity

Compute the integral:

$$\int {\tan }^{3}x{\sec }^{2}xdx$$

(a) Using $du={\sec }^{2}xdx$.

(b) Using $du=\sec x\tan xdx$.

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Solution

Solutions---0040-05

03 $★$

06

Power product with negative power

Compute the integral:

$$\int \sin 7x{\sec }^{5}7xdx$$

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Solution

Solutions---0040-06

Trig substitution

04 $★$

03

Trig sub

Compute the integral:

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

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Solution

Solutions---0050-03

05 $★$

04

Trig sub

Compute the integral:

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

Hint: complete the square and then substitute.

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Solution

Solutions---0050-04

06

05

Trig sub

Compute the integral:

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

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Solution

Solutions---0050-05

07

06

Double sub: $u$-sub then trig sub

Compute the definite integral:

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

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Solution

Solutions---0050-06

08

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 $(x_1,x_2)=(-\mathrm{15,15})$.

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Solution

Solutions---0050-07