Integration Question Need Help (1 Viewer)

chris_power_96 · Apr 26, 2013

How do you solve the integral of cos^(5)x/(cos^(5)x+sin^(5)x) between pi/2 and 0

RealiseNothing · Apr 26, 2013

Let $I = \int_{0}^{\frac{\pi}{2}}\frac{cos^5x}{cos^5x+sin^5x}dx$

Using the property that:

$\int_{0}^{a} f(x)dx = \int_{0}^{a} f(a-x)dx$

$I = \int_{0}^{\frac{\pi}{2}} \frac{cos^5(\frac{\pi}{2}-x)}{cos^5(\frac{\pi}{2}-x)+sin^5(\frac{\pi}{2}-x)} dx$

$I = \int_{0}^{\frac{\pi}{2}} \frac{sin^5x}{sin^5x+cos^5x}dx$

Adding this to the original integral gives:

$2I = \int_{0}^{\frac{\pi}{2}} \frac{cos^5x+sin^5x}{cos^5x+sin^5x}dx$

$2I = \int_{0}^{\frac{\pi}{2}} 1dx$

$2I = \frac{\pi}{2}$

$I = \frac{\pi}{4}$

Therefore:

$\int_{0}^{\frac{\pi}{2}}\frac{cos^5x}{cos^5x+sin^5x}dx = \frac{\pi}{4}$

Integration Question Need Help (1 Viewer)

chris_power_96

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RealiseNothing

what is that?It is Cowpea

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