What is the integral from 0 to pi/2 of cos^6(x) ?

The integral from 0 to pi/2 of cos^6(x) is (5pi)/(32)

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integral from 0 to pi/2 of cos^6(x)

\int_{0}^{\frac{π}{2}} cos^{6} (x) d x

\frac{5 π}{32}

Decimal Notation

0.49087 \dots

Solution steps

\int_{0}^{\frac{π}{2}} cos^{6} (x) d x

Apply Integral Reduction

= [\frac{sin ( x ) cos ^{5} ( x )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} \cdot \int_{0}^{\frac{π}{2}} cos^{4} (x) d x

\int_{0}^{\frac{π}{2}} cos^{4} (x) d x = \frac{3 π}{16}

= [\frac{sin ( x ) cos ^{5} ( x )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{16}

Simplify

[\frac{sin ( x ) cos ^{5} ( x )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{16} : [\frac{1}{6} cos^{5} (x) sin (x)]_{0}^{\frac{π}{2}} + \frac{5 π}{32}

= [\frac{1}{6} cos^{5} (x) sin (x)]_{0}^{\frac{π}{2}} + \frac{5 π}{32}

Compute the boundaries:

0

= 0 + \frac{5 π}{32}

Simplify

= \frac{5 π}{32}

\frac{d}{d x} (- sin (5 x) \cdot 5)

\int sec^{2} (x) tan (x) e^{s e c (x)} d x

sin (cos (5 x))

\int sin (x) + 8 d x

\int \frac{8}{x ^{8}} d x

What is the integral from 0 to pi/2 of cos^6(x) ?
The integral from 0 to pi/2 of cos^6(x) is (5pi)/(32)