1. ## Two integrations

Hi all,

Got the good old exam tomorrow and I'm running through past papers. These two questions are giving me a few problems and unfortunately being multiple choice I don't have worked examples.

Methods we've learnt basically sum up into substitution, partial fractions, t-substitution, integration by parts. If you could solve the problems using these methods it would be greatly appreciated.

Here are the two.

$\displaystyle \int \cos(\ln(x)) dx$

$\displaystyle \int \sin^2 x \cos 3x dx$

2. integrate sin(x)^2 *cos(3x) - Wolfram|Alpha
wolframalpha will integrate for you and then you can click show steps in the right corner, and it will give you a step by step runthrough of how it got the answer

3. $\displaystyle \int \cos(\ln{x}) \, dx$

$\displaystyle u = \ln{x}$

$\displaystyle x = e^u$

$\displaystyle dx = e^u \, du$

$\displaystyle \int \cos{u} \cdot e^u \, du$

use parts

4. $\displaystyle \int \cos(\ln(x)) dx$

Try parts.

Let $\displaystyle u=cos(ln(x)), \;\ du=\frac{-sin(ln(x))}{x}dx, \;\ dv=dx, \;\ v=x$

We get:

$\displaystyle xcos(ln(x))+\int sin(ln(x))dx$

Now, do it again:

$\displaystyle u=sin(ln(x)), \;\ du=\frac{cos(ln(x))}{x}dx, \;\ v=x, \;\ dv=dx$

$\displaystyle \int cos(ln(x))dx=xcos(ln(x))+xsin(ln(x))-\int cos(ln(x))dx$

See, we're going in circles. Now, add the integral on the right side to both sides:

$\displaystyle 2\int cos(ln(x))dx=xcos(ln(x))+xsin(ln(x))$

Divide by 2:

$\displaystyle \int cos(ln(x))dx=\frac{xcos(ln(x))}{2}+\frac{xsin(ln(x ))}{2}$