Good examples on Integrals

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Aug 9th 2007, 07:59 AM
wizzler

Good examples on Integrals

Hey!

Can anyone give me good examples on how to solve integrals like:

$\int_{0}^{\pi/2} \frac{cos x}{sin^2x + sin^3 x} dx$

with partial integration/substitution methods?
Aug 9th 2007, 08:11 AM
Krizalid

First rewrite the integrand as follows

$\int_0^{\pi/2}\frac{\cos x}{\sin^2x+\sin^3x}\,dx=\int_0^{\pi/2}\frac{\cos x}{\sin^2x(1+\sin x)}\,dx$

Then set $u=1+\sin x\implies du=\cos x\,dx$ , which yields

$\int_1^2\frac1{u(u-1)^2}\,du$

Just post if you cannot take it from there.
Aug 9th 2007, 08:14 AM
ThePerfectHacker

Just be careful this is a "Improper Integral of the 2nd Type".
Aug 9th 2007, 08:34 AM
wizzler

Quote:

Originally Posted by Krizalid

First rewrite the integrand as follows

$\int_0^{\pi/2}\frac{\cos x}{\sin^2x+\sin^3x}\,dx=\int_0^{\pi/2}\frac{\cos x}{\sin^2x(1+\sin x)}\,dx$

Then set $u=1+\sin x\implies du=\cos x\,dx$ , which yields

$\int_1^2\frac1{u(u-1)^2}\,du$

Just post if you cannot take it from there.

so $\int_1^2\frac1{u(u-1)^2}\,du$ can be put as

$u \int_1^2\frac1{(u-1)^2}\,du$ and $\frac1{(u-1)^2}\$ is $ln((u-1)^2)$ or am i thinking it totally wrong now?
Aug 9th 2007, 08:40 AM
wizzler

neuu of course...

i part integrate it so i get $\frac{a}{u^2} + \frac{b}{u} + \frac{c}{u + 1}$
Aug 9th 2007, 09:05 AM
Krizalid

Quote:

Originally Posted by wizzler

so $\int_1^2\frac1{u(u-1)^2}\,du$ can be put as

$u \int_1^2\frac1{(u-1)^2}\,du$

This step is not correct 'cause "u" it's a function of "x", not a constant.

We have

$<br /> \begin{aligned}<br /> \frac1{u(u-1)^2}&=\frac{u^2-2u+1+u-u^2+u}{u(u-1)^2}\\&=\frac{(u-1)^2+u-u(u-1)}{u(u-1)^2}\\&=\frac1u+\frac1{(u-1)^2}-\frac1{u-1}<br /> \end{aligned}<br />$

Kill it now.

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