Find out the answer to "I have to calculate this Integral -∫0π2sin72xsin72x+cos72xdx"

ashes86047xhz

Answered question

2022-03-29

I have to calculate this Integral -

\int_{0}^{\frac{π}{2}} \frac{\sin^{\frac{7}{2}} x}{\sin^{\frac{7}{2}} x + \cos^{\frac{7}{2}} x} d x

Korbin Rivera

Beginner2022-03-30Added 11 answers

Use the following:

\int_{a}^{b} f (x), d x = \int_{a}^{b} f (a + b - x), d x

This property can be proven easily by substituting

a + b - x = t

Now consider

f (x) = \frac{\sin^{\frac{7}{2}} x}{\sin^{\frac{7}{2}} x + \cos^{\frac{7}{2}} x}

and

a = 0, b = \frac{π}{2}

, to get

I = \int_{0}^{\frac{π}{2}} \frac{\sin^{\frac{7}{2}} x}{\sin^{\frac{7}{2}} x + \cos^{\frac{7}{2}} x} d x = \int_{0}^{\frac{π}{2}} \frac{\sin^{\frac{7}{2}} (\frac{π}{2} - x)}{\sin^{\frac{7}{2}} (\frac{π}{2} - x) + \cos^{\frac{7}{2}} (\frac{π}{2} - x)} d x

Thus we have

2 I = \int_{0}^{\frac{π}{2}} : \frac{\sin^{\frac{7}{2}} x + \cos^{\frac{7}{2}} x}{\sin^{\frac{7}{2}} x + \cos^{\frac{7}{2}} x} d x = \int_{0}^{\frac{π}{2}} 1, d x = \frac{π}{2} .

Thus

I = \frac{π}{4}

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