What is the integral from 0 to pi of cos^8(x/2) ?

The integral from 0 to pi of cos^8(x/2) is (35pi)/(128)

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integral from 0 to pi of cos^8(x/2)

\int_{0}^{π} cos^{8} (\frac{x}{2}) d x

\frac{35 π}{128}

Solution steps

\int_{0}^{π} cos^{8} (\frac{x}{2}) d x

Apply u-substitution

: \int_{0}^{\frac{π}{2}} cos^{8} (u) \cdot 2 d u

= \int_{0}^{\frac{π}{2}} cos^{8} (u) \cdot 2 d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{0}^{\frac{π}{2}} cos^{8} (u) d u

Apply Integral Reduction

= 2 ([\frac{sin ( u ) cos ^{7} ( u )}{8}]_{0}^{\frac{π}{2}} + \frac{7}{8} \cdot \int_{0}^{\frac{π}{2}} cos^{6} (u) d u)

\int_{0}^{\frac{π}{2}} cos^{6} (u) d u = \frac{5 π}{32}

= 2 ([\frac{sin ( u ) cos ^{7} ( u )}{8}]_{0}^{\frac{π}{2}} + \frac{7}{8} \cdot \frac{5 π}{32})

Simplify

2 ([\frac{sin ( u ) cos ^{7} ( u )}{8}]_{0}^{\frac{π}{2}} + \frac{7}{8} \cdot \frac{5 π}{32}) : 2 ([\frac{1}{8} cos^{7} (u) sin (u)]_{0}^{\frac{π}{2}} + \frac{35 π}{256})

= 2 ([\frac{1}{8} cos^{7} (u) sin (u)]_{0}^{\frac{π}{2}} + \frac{35 π}{256})

Compute the boundaries:

0

= 2 (0 + \frac{35 π}{256})

Simplify

= \frac{35 π}{128}

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What is the integral from 0 to pi of cos^8(x/2) ?
The integral from 0 to pi of cos^8(x/2) is (35pi)/(128)