integral

**kittycat** · August 3rd 2007, 06:18 AM

-1/2 times the integral (from 0 to pi/2) of cos^4 theta - 2cos^2 theta d(theta)

Thank you very much.

**tukeywilliams** · August 3rd 2007, 06:42 AM

$-\frac{1}{2} \int_{0}^{\pi/2} \cos^{4} \theta - 2 \cos^{2} \theta \ d \theta$

Use power reduction formulas: $\sin^{2} \theta = \frac{1- \cos 2 \theta}{2}$

and $\cos^{2} \theta = \frac{1+ \cos 2 \theta}{2}$

**kittycat** · August 3rd 2007, 06:49 AM

hey turkeywilliam,

I did. But I just can't get the right answer. I got (5pi)/64 +1/8 . But the correct answer is (5pi)/32.

Don't know why?

**topsquark** · August 3rd 2007, 07:07 AM

Originally Posted by tukeywilliams

$-\frac{1}{2} \int_{0}^{\pi/2} \cos^{4} \theta - 2 \cos^{2} \theta \ d \theta$

Use power reduction formulas: $\sin^{2} \theta = \frac{1- \cos 2 \theta}{2}$

and $\cos^{2} \theta = \frac{1+ \cos 2 \theta}{2}$

Originally Posted by kittycat

hey turkeywilliam,

I did. But I just can't get the right answer. I got (5pi)/64 +1/8 . But the correct answer is (5pi)/32.

Don't know why?

$-\frac{1}{2} \int_{0}^{\pi/2}( (cos^2( \theta ) )^2 - 2 cos^2( \theta )) d \theta$

$= -\frac{1}{2} \int_{0}^{\pi/2}\left ( \left ( \frac{1 + cos(2 \theta )}{2} \right )^2 - 2 \left ( \frac{ 1 + cos(2 \theta )}{2} \right ) \right ) d \theta$

$= -\frac{1}{8} \int_{0}^{\pi/2} (1 + cos(2 \theta ))^2 d \theta + \frac{1}{2} \int_0^{\pi/2} (1 + cos(2 \theta ) ) d \theta$

Expand the integrand in the first integral:
$= -\frac{1}{8} \int_{0}^{\pi/2}(1 + 2cos(2 \theta ) + cos^2(2 \theta) ) d \theta + \frac{1}{2} \int_0^{\pi/2} (1 + cos(2 \theta ) ) d \theta$

$= \frac{3}{8} \int_{0}^{\pi/2} d \theta + \frac{1}{4} \int_0^{\pi/2} cos(2 \theta ) d \theta - \frac{1}{8} \int_0^{\pi/2} cos^2(2 \theta) d \theta$

Now use power reduction in the last integral again:
$= \frac{3}{8} \int_{0}^{\pi/2} d \theta + \frac{1}{4} \int_0^{\pi/2} cos(2 \theta ) d \theta - \frac{1}{8} \int_0^{\pi/2} \frac{1 + cos(4 \theta)}{2} d \theta$

$= \frac{5}{16} \int_{0}^{\pi/2} d \theta + \frac{1}{4} \int_0^{\pi/2} cos(2 \theta ) d \theta - \frac{1}{16} \int_0^{\pi/2} cos(4 \theta) d \theta$

Now try it.

-Dan

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