1. ## Easy Integrals!!

Evaluate the following integrals

2. 1st one comes out to:
$\displaystyle \frac{1}{4}x^2 (2log(x) - 1)$

2nd one comes out to:
$\displaystyle \frac{1}{5}(2log(x - 2) + 3log(x + 3))$

I think! not 100% certain there.

3. Originally Posted by matty888
Evaluate the following integrals

for the first one use the change of base formula to get

$\displaystyle \int_{1}^{2}x\log(x)dx=\int_{1}^{2}x\frac{\ln(x)}{ \ln(10)}dx=\frac{1}{\ln(10)}\int_{1}^{2}x\ln(x)dx$

use integration by parts with u=ln(x) dv=x

That should get you started

4. Hi
Originally Posted by matty888
Let's find the partial fraction decomposition of $\displaystyle \frac{x}{x^2+x-6}$.

The discriminant of the denominator is $\displaystyle \Delta = 1^2-4\cdot1\cdot (-6)=25>0$ hence the denominator has to different real roots : $\displaystyle \frac{-1\pm\sqrt{25}}{2}$ that is to say -3 and 2. Hence $\displaystyle x^2+x-6=(x+3)(x-2)$

It gives $\displaystyle \frac{x}{x^2+x-6} = \frac{x}{(x+3)(x-2)}=\frac{\alpha}{x+3}+\frac{\beta}{x-2}$ which has to be solved for $\displaystyle \alpha$ and $\displaystyle \beta$.

Can you take it from here ?

5. \displaystyle \begin{aligned} \frac{x}{x^{2}+x-6}&=\frac{2x+1-1}{2(x-2)(x+3)} \\ & =\frac{1}{2}\left[ \frac{2x+1}{x^{2}+x-6}-\frac{(x+3)-(x-2)}{5(x-2)(x+3)} \right] \\ & =\frac{1}{2}\left[ \frac{2x+1}{x^{2}+x-6}-\frac{1}{5}\left\{ \frac{1}{x-2}-\frac{1}{x+3} \right\} \right]. \end{aligned}

Integrate.

P.S.: first integral can be tackled by double integration technique too.

6. Originally Posted by Krizalid
\displaystyle \begin{aligned} \frac{x}{x^{2}+x-6}&=\frac{2x+1-1}{2(x-2)(x+3)} \\ & =\frac{1}{2}\left[ \frac{2x+1}{x^{2}+x-6}-\frac{(x+3)-(x-2)}{5(x-2)(x+3)} \right] \\ & =\frac{1}{2}\left[ \frac{2x+1}{x^{2}+x-6}-\frac{1}{5}\left\{ \frac{1}{x-2}-\frac{1}{x+3} \right\} \right]. \end{aligned}

Integrate.
Haha! i LOVE seeing when you do that!