Evaluate the definite Integral

**zorro** · Dec 5th 2009, 12:51 AM

Question : Evaluate

$ \int_{-1}^{1} \ \frac{3 + sin^3 x}{1 + x^2} \ dx $

**flyingsquirrel** · Dec 5th 2009, 01:08 AM

Originally Posted by zorro

Question : Evaluate

$ \int_{-1}^{1} \ \frac{3 + sin^3 x}{1 + x^2} \ dx $

$ \int_{-1}^{1} \frac{3 + \sin^3 x}{1 + x^2} \ \mathrm{d}x=3\int_{-1}^{1} \frac{1}{1 + x^2} \ \mathrm{d}x+\int_{-1}^{1} \frac{\sin^3 x}{1 + x^2} \ \mathrm{d}x $

Hint : note that $x\mapsto \frac{\sin^3 x}{1 + x^2}$ is odd.

**zorro** · Dec 8th 2009, 12:27 AM

Originally Posted by flyingsquirrel

$ \int_{-1}^{1} \frac{3 + \sin^3 x}{1 + x^2} \ \mathrm{d}x=3\int_{-1}^{1} \frac{1}{1 + x^2} \ \mathrm{d}x+\int_{-1}^{1} \frac{\sin^3 x}{1 + x^2} \ \mathrm{d}x $

Hint : note that $x\mapsto \frac{\sin^3 x}{1 + x^2}$ is odd.

I am able to integrate the first term but not the last one

= $3 \int_{-1}^{1} \frac{1}{1 + x^2} + \int_{-1}^{1} \frac{sin^3 x}{1 + x^2}$

= $I_1 + I_2$

$I_1$ = $3 \left[ tan^{-1} (1) - tan^{-1} (-1) \right] + c_1$

= $3 \left[ tan^{-1} (1) - tan^{-1} (-1) \right] + c_1$

= $3 \left[ \frac{\pi}{4} +\frac{\pi}{4} \right] + c_1$

= $3 \left[ \frac{2 \pi}{4} \right] + c_1$

= $\left[ \frac{6 \pi}{4} \right] + c_1$

$I_2$ = $\int_{-1}^{1} \frac{sin^3 x}{1 + x^2}$ $dx_2$

= $\int_{-1}^{1} \frac{sin^3 x}{1 + x^2}$ $dx_2$

I am stuck here!!!!!......

**dedust** · Dec 8th 2009, 01:59 AM

Originally Posted by flyingsquirrel

Hint : note that $x\mapsto \frac{\sin^3 x}{1 + x^2}$ is odd.

so $\int_{-1}^{1} \frac{\sin^3 x}{1 + x^2} \ \mathrm{d}x = 0$

**flyingsquirrel** · Dec 8th 2009, 02:03 AM

Originally Posted by zorro

$I_1$ = $3 \left[ tan^{-1} (1) - tan^{-1} (-1) \right] + c_1$

There is no need to add a constant since $\int_{-1}^1\frac{1}{1+x^2}\,\mathrm{d}x$ is a definite integral.

Originally Posted by zorro

= $\left[ \frac{6 \pi}{4} \right] + c_1$

$\frac{6\pi}{4}=\frac{3\times 2\pi}{2\times 2}=\frac{3\pi}{2}$

Originally Posted by zorro

$I_2$ = $\int_{-1}^{1} \frac{sin^3 x}{1 + x^2}$ $dx_2$

$I_2 = \int_{-1}^0\frac{\sin^3x}{1+x^2}\,\mathrm{d}x + \int_0^1\frac{\sin^3x}{1+x^2}\,\mathrm{d}x$

Substitute $x=-t$ ( $\implies t=-x$ and $\mathrm{d}t=-\mathrm{d}x$ ) in the first integral :

$\int_{-1}^0\frac{\sin^3x}{1+x^2}\,\mathrm{d}x =-\int_{1}^0\frac{\sin^3(-t)}{1+(-t)^2}\,\mathrm{d}t=\int_0^1\frac{-\sin^3t}{1+t^2}\,\mathrm{d}t=-\int_0^1\frac{\sin^3t}{1+t^2}\,\mathrm{d}t$

Thus

$I_2 = -\int_0^1\frac{\sin^3t}{1+t^2}\,\mathrm{d}t+\int_0^ 1\frac{\sin^3x}{1+x^2}\,\mathrm{d}x=0.$

This is a special case of a more general result: Let $f:\mathbb{R}\to\mathbb{R}$ be an odd function and let $A$ be a real number. Then

$\int_{-A}^Af(t)\,\mathrm{d}t=0.$

Edit :

Originally Posted by dedust

so $\int_{-1}^{1} \frac{\sin^3 x}{1 + x^2} \ \mathrm{d}x = 0$

Yes !

**zorro** · Dec 8th 2009, 02:17 AM

Thank u all very much for helping me

**Krizalid** · Dec 8th 2009, 05:48 AM

as for that property about the odd function, we also require that $f$ must be integrable there.

**zorro** · Dec 8th 2009, 12:01 PM

Originally Posted by Krizalid

as for that property about the odd function, we also require that $f$ must be integrable there.

could u please explain it in non mathematical terms ,as i m not that bright??

**zorro** · Dec 21st 2009, 03:30 PM

Originally Posted by zorro

could u please explain it in non mathematical terms ,as i m not that bright??

Thanks u all for helping

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Evaluate the definite Integral

Stuck again !!!

Thanks u very much

what do u mean by that

Got it

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