1. ## Integration Problem

Evaluate
$\displaystyle \int\limits_{0}^{\pi} \frac{xsinx}{1+sinx}dx.$

2. Set $\displaystyle u=\pi-x$ then $\displaystyle I = \int_0^\pi {\tfrac{{x \cdot \sin \left( x \right)}} {{1 + \sin \left( x \right)}}dx} = \int_0^\pi {\tfrac{{\left( {\pi - x} \right) \cdot \sin \left( {\pi - x} \right)}} {{1 + \sin \left( {\pi - x} \right)}}dx} = \int_0^\pi {\tfrac{{\left( {\pi - x} \right) \cdot \sin \left( x \right)}} {{1 + \sin \left( x \right)}}dx}$

So: $\displaystyle 2 \cdot I = \int_0^\pi {\tfrac{{x \cdot \sin \left( x \right)}} {{1 + \sin \left( x \right)}}dx} + \int_0^\pi {\tfrac{{\left( {\pi - x} \right) \cdot \sin \left( x \right)}} {{1 + \sin \left( x \right)}}dx} = \int_0^\pi {\left( {\tfrac{{x \cdot \sin \left( x \right)}} {{1 + \sin \left( x \right)}} + \tfrac{{\left( {\pi - x} \right) \cdot \sin \left( x \right)}} {{1 + \sin \left( x \right)}}} \right)dx}$

We get $\displaystyle 2 \cdot I = \pi \cdot \int_0^\pi {\left( {\tfrac{{\sin \left( x \right)}} {{1 + \sin \left( x \right)}}} \right)dx}$ and this can be solved by standard methods

3. ## unable to understand ..

I am unable to understand hw u carried out the steps.................

4. Originally Posted by varunnayudu
I am unable to understand hw u carried out the steps.................
Read the reply again - carefully. Now try and be more specific - what part or parts don't you understand?

5. And kindly show your try

6. I'm also having trouble following.

Where did u come into it? i.e. how did x become $\displaystyle \pi - x$?

7. Originally Posted by Prove It
I'm also having trouble following.

Where did u come into it? i.e. how did x become $\displaystyle \pi - x$?
The substitution $\displaystyle u = \pi - x$ is made and then the dummy variable is switched back to x:

$\displaystyle I = \int_0^\pi {\frac{{x \cdot \sin \left( x \right)}} {{1 + \sin \left( x \right)}} \, dx}$

Substitute $\displaystyle u = \pi - x$:

$\displaystyle = \int_\pi^0 {\frac{{\left( {\pi - u} \right) \cdot \sin \left( {\pi - u} \right)}}{{1 + \sin \left( {\pi - u} \right)}} \, (-du) }$

$\displaystyle = \int_0^\pi {\frac{{\left( {\pi - u} \right) \cdot \sin \left( {\pi - u} \right)}}{{1 + \sin \left( {\pi - u} \right)}} \, du }$

Now switch the dummy variable back to x:

$\displaystyle = \int_0^\pi {\frac{{\left( {\pi - x} \right) \cdot \sin \left( {\pi - x} \right)}} {{1 + \sin \left( {\pi - x} \right)}} \, dx}$

etc.

This is a standard technique.

8. ## Still hve a problem

Now switch the dummy variable back to x:

---------------------------------------------

Hw can one substitute $\displaystyle (\pi - u)$ into $\displaystyle (\pi - x)$

and secondly even after doing the sub the sum is not evaluated it is still the same ...........only instead of $\displaystyle x$ u have $\displaystyle (\pi - x)$

$\displaystyle I = \int_0^\pi {\tfrac{{x \cdot \sin \left( x \right)}} {{1 + \sin \left( x \right)}}dx} = \int_0^\pi {\tfrac{{\left( {\pi - x} \right) \cdot \sin \left( {\pi - x} \right)}} {{1 + \sin \left( {\pi - x} \right)}}dx} = \int_0^\pi {\tfrac{{\left( {\pi - x} \right) \cdot \sin \left( x \right)}} {{1 + \sin \left( x \right)}}dx}$

$\displaystyle 2 \cdot I = \int_0^\pi {\tfrac{{x \cdot \sin \left( x \right)}} {{1 + \sin \left( x \right)}}dx} + \int_0^\pi {\tfrac{{\left( {\pi - x} \right) \cdot \sin \left( x \right)}} {{1 + \sin \left( x \right)}}dx}$

tell me hw u got this conversion ..............

9. Hi Varun,
$\displaystyle I=\int^{a}_{b}{f(x)}$

$\displaystyle I=\int^{a}_{b}{f(a+b-x)}$

I hope you know the above property
Here$\displaystyle f(x)=\frac{xsinx}{1-sin(x)}$
Hence
the two equations are

$\displaystyle I+I= 2I$
which is given

----------------------------------------------------------------
to prove that property
Put t=a+b-x
dt=-dx
When x=a ,t=b
and when x=b,t=a
Now
$\displaystyle \int^{a}_{b}{f(x)dx}=-\int^{b}_{a}{f(a+b-t)dt}$$\displaystyle =\int^{a}_{b}{f(a+b-t)dt}$
As we know
$\displaystyle \int^{x}_{y}{g(\alpha)d{\alpha }} =\int^{x}_{y}{g(\beta)d{\beta }}$
So$\displaystyle =\int^{a}_{b}{f(a+b-x)dx}=\int^{a}_{b}{f(x)dx}$
And that's the end of that show..

10. Originally Posted by varunnayudu

Now switch the dummy variable back to x:

---------------------------------------------

Hw can one substitute $\displaystyle (\pi - u)$ into $\displaystyle (\pi - x)$

and secondly even after doing the sub the sum is not evaluated it is still the same ...........only instead of $\displaystyle x$ u have $\displaystyle (\pi - x)$

$\displaystyle I = \int_0^\pi {\tfrac{{x \cdot \sin \left( x \right)}} {{1 + \sin \left( x \right)}}dx} = \int_0^\pi {\tfrac{{\left( {\pi - x} \right) \cdot \sin \left( {\pi - x} \right)}} {{1 + \sin \left( {\pi - x} \right)}}dx} = \int_0^\pi {\tfrac{{\left( {\pi - x} \right) \cdot \sin \left( x \right)}} {{1 + \sin \left( x \right)}}dx}$

$\displaystyle 2 \cdot I = \int_0^\pi {\tfrac{{x \cdot \sin \left( x \right)}} {{1 + \sin \left( x \right)}}dx} + \int_0^\pi {\tfrac{{\left( {\pi - x} \right) \cdot \sin \left( x \right)}} {{1 + \sin \left( x \right)}}dx}$

tell me hw u got this conversion ..............
Maybe I can help. After you make the substitution $\displaystyle x = \pi - u$ your integral, as Mr. F says is

$\displaystyle I = \int_{\pi}^0 \frac{(\pi-u) \sin (\pi-u)}{1 + \sin(\pi-u)}(-du)$

using the fact that

$\displaystyle \sin(\pi-u) = \sin u$

then (I used the negative to switch the limits of integration)

$\displaystyle I = \int_0^{\pi} \frac{(\pi-u) \sin (u)}{1 + \sin(u)}\,du$

or, upon expanding,

$\displaystyle I = \int_0^{\pi} \frac{\pi \sin (u)}{1 + \sin(u)}\,du - \int_0^{\pi} \frac{u \sin (u)}{1 + \sin(u)}\,du$

If you look at your original integral and the second one of the right these are the same except for the variable $\displaystyle u$. As this is only a dummy variable (it is used and never appears in the answer) then we can replace $\displaystyle u$ with $\displaystyle x$ giving

$\displaystyle I = \int_0^{\pi} \frac{\pi \sin (u)}{1 + \sin(u)}\,du - I$

or

$\displaystyle 2I = \int_0^{\pi} \frac{\pi \sin (u)}{1 + \sin(u)}\,du$

as others have said. Others can correct me if I'm wrong but I believe this question appears in Stewart's Calculus book.