# Math Help - The Substitution Rule

1. ## The Substitution Rule

Can someone help me with these two problems. Thanks a lot.
1. integral of cos^3 thelta sin thelta d thelta, u= cos thelta
this is what I did so far: du/dx= -sin thelta du= -sin thelta dx => -du=sin thelta d thelta ...I dont even know if this is correct. I need to sub. u into the function but dont know how.
2. integral sec^2(1/x)/x^2 dx , u = 1/x
I got -ln x du = dx .....I dont know what to do at all

2. $\int \cos^3{t} \cdot \sin{t} \, dt$

$u = \cos{t}$

$du = -\sin{t} \, dt$

$-\int \cos^3{t} \cdot (-\sin{t}) \, dt$

sub ...

$-\int u^3 \, du$

do the integration.

$\int \sec^2\left(\frac{1}{x}\right) \cdot \frac{1}{x^2} \, dx$

$u = \frac{1}{x}$

$du = -\frac{1}{x^2} \, dx$

$-\int \sec^2\left(\frac{1}{x}\right) \cdot \left(-\frac{1}{x^2}\right) \, dx$

sub ...

$-\int \sec^2{u} \, du$

do the integration.

3. Here's exactly what you do for the first one so hopefully you can get the method down.

$\int\cos^3\theta\sin\theta d\theta$. So if you set $u=\cos\theta, \frac{du}{d\theta}=-\sin\theta$. So move over the $d\theta$, $du=-\sin\theta d\theta$. Solve for $d\theta$, $d\theta=-\frac{du}{\sin\theta}$. Substituting this in,

$\int\cos^3\theta\sin\theta d\theta=\int u^3\frac{-\sin\theta du}{\sin\theta}=-\int u^3 du=-\frac{1}{4}u^4 + C = -\frac{1}{4}\cos^4\theta + C.$

This follows after you remember what we defined u as.