# Definite Integration.

• Dec 23rd 2008, 01:39 AM
varunnayudu
Definite Integration.
Evaluate

$\displaystyle I = \int_{0}^{\pi} \frac{x sin x}{1+cos^2 x} dx$
• Dec 23rd 2008, 02:07 AM
Moo
Hello,
Quote:

Originally Posted by varunnayudu
Evaluate

$\displaystyle I = \int_{0}^{\pi} \frac{x sin x}{1+cos^2 x} dx$

Let $\displaystyle t=\pi-x$ :

$\displaystyle I=-\int_\pi^0 \frac{(\pi-t) \sin(\pi-t)}{1+\cos^2(\pi-t)} ~ dt=\int_0^\pi \frac{(\pi-t) \sin(\pi-t)}{1+\cos^2(\pi-t)} ~ dt$

But $\displaystyle \sin(\pi-t)=\sin(t)$ and $\displaystyle \cos(\pi-t)=-\cos(t)$
Hence :

$\displaystyle I=\int_0^\pi \frac{(\pi-t) \sin(t)}{1+\cos^2(t)}~dt$

$\displaystyle I=\pi \int_0^\pi \frac{\sin(t)}{1+\cos^2(t)} ~dt-\underbrace{\int_0^\pi \frac{t \sin(t)}{1+\cos^2(t)} ~ dt}_{=I}$

$\displaystyle 2I=\pi \int_0^\pi \frac{\sin(t)}{1+\cos^2(t)} ~ dt$

$\displaystyle I=\frac{\pi}{2} \int_0^\pi \frac{\sin(t)}{1+\cos^2(t)} ~ dt$

Now substitute $\displaystyle \varphi=\cos(t)$ and it'll be easy.
• Dec 30th 2008, 09:02 PM
zorro
hw did u do that
hw did u get that
• Dec 30th 2008, 11:08 PM
mr fantastic
Quote:

Originally Posted by zorro
hw did u get that

You will need to be more specific. What part of Moo's reply don't you understand?
• Jan 1st 2009, 01:15 AM
zorro
Please tell me if its correct.
Quote:

Originally Posted by Moo
Hello,

Let $\displaystyle t=\pi-x$ :

$\displaystyle I=-\int_\pi^0 \frac{(\pi-t) \sin(\pi-t)}{1+\cos^2(\pi-t)} ~ dt=\int_0^\pi \frac{(\pi-t) \sin(\pi-t)}{1+\cos^2(\pi-t)} ~ dt$

But $\displaystyle \sin(\pi-t)=\sin(t)$ and $\displaystyle \cos(\pi-t)=-\cos(t)$
Hence :

$\displaystyle I=\int_0^\pi \frac{(\pi-t) \sin(t)}{1+\cos^2(t)}~dt$

$\displaystyle I=\pi \int_0^\pi \frac{\sin(t)}{1+\cos^2(t)} ~dt-\underbrace{\int_0^\pi \frac{t \sin(t)}{1+\cos^2(t)} ~ dt}_{=I}$

$\displaystyle 2I=\pi \int_0^\pi \frac{\sin(t)}{1+\cos^2(t)} ~ dt$

$\displaystyle I=\frac{\pi}{2} \int_0^\pi \frac{\sin(t)}{1+\cos^2(t)} ~ dt$

Now substitute $\displaystyle \varphi=\cos(t)$ and it'll be easy.

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$\displaystyle cos(t) = \varphi$
therefore
$\displaystyle - sin t dt= d\varphi$

$\displaystyle - \frac{\pi}{2} \int_0^\pi \frac{1}{1+\varphi^2}d\varphi$

what to do next ................(Wait)
• Jan 1st 2009, 01:49 AM
mr fantastic
Quote:

Originally Posted by zorro
--------------------------------------------------------------------

$\displaystyle cos(t) = \varphi$
therefore
$\displaystyle - sin t dt= d\varphi$

$\displaystyle - \frac{\pi}{2} \int_0^\pi \frac{1}{1+\varphi^2}d\varphi$

what to do next ................(Wait)

$\displaystyle \int \frac{1}{1+\varphi^2} \, d\varphi$ is a standard form and is equal to $\displaystyle \tan^{-1} \varphi$ (I have omitted the arbitrary cnostant of integration).
• Jan 1st 2009, 02:08 AM
zorro
Quote:

Originally Posted by mr fantastic
$\displaystyle \int \frac{1}{1+\varphi^2} \, d\varphi$ is a standard form and is equal to $\displaystyle \tan^{-1} \varphi$ (I have omitted the arbitrary cnostant of integration).

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= -$\displaystyle \frac{\pi}{2} (tan^{-1} \pi - tan^{-1}0)$

= - $\displaystyle \frac{\pi}{2} (0-0)$

= - $\displaystyle \frac{\pi}{2} .0$

= $\displaystyle 0$

is this right .........(Wait)
• Jan 1st 2009, 02:10 AM
Chop Suey
You forgot to change the limits.
• Jan 1st 2009, 05:00 AM
tom@ballooncalculus
http://www.ballooncalculus.org/defPi.png

The straight dashed line diff's / anti-diff's with respect to the dashed balloon expression.

Don't integrate - balloontegrate! http://www.ballooncalculus.org/examples]
• Jan 4th 2009, 06:39 PM
zorro
hw did t conert to pi
Quote:

Originally Posted by tom@ballooncalculus
http://www.ballooncalculus.org/defPi.png

The straight dashed line diff's / anti-diff's with respect to the dashed balloon expression.

Don't integrate - balloontegrate! http://www.ballooncalculus.org/examples]

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Here hw did t=$\displaystyle \pi$ turn into 1 and -1.
• Jan 4th 2009, 06:56 PM
mr fantastic
Quote:

Originally Posted by zorro
-------------------------------------------------------------------

Here hw did t=$\displaystyle \pi$ turn into 1 and -1.

Recall that there was a change of variable: $\displaystyle \varphi = \cos t$.

$\displaystyle t = 0 \Rightarrow \varphi = \cos 0 = 1$.

$\displaystyle t = \pi \Rightarrow \varphi = \cos \pi = -1$.
• Jan 4th 2009, 07:10 PM
qpmathelp
Quote:

Originally Posted by zorro
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Here hw did t=$\displaystyle \pi$ turn into 1 and -1.

they are the limits for the new variable