What is the integral from 0 to 1/2 of cos^3(pix) ?

The integral from 0 to 1/2 of cos^3(pix) is 2/(3pi)

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integral from 0 to 1/2 of cos^3(pix)

\int_{0}^{\frac{1}{2}} cos^{3} (π x) d x

\frac{2}{3 π}

Decimal

0.21220 \dots

Solution steps

\int_{0}^{\frac{1}{2}} cos^{3} (π x) d x

Apply u-substitution

: \int_{0}^{\frac{π}{2}} cos^{3} (u) \frac{1}{π} d u

= \int_{0}^{\frac{π}{2}} cos^{3} (u) \frac{1}{π} d u

Take the constant out:

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

= \frac{1}{π} \cdot \int_{0}^{\frac{π}{2}} cos^{3} (u) d u

Rewrite using trig identities

= \frac{1}{π} \cdot \int_{0}^{\frac{π}{2}} (1 - sin^{2} (u)) cos (u) d u

Apply u-substitution

: \int_{0}^{1} 1 - v^{2} d v

= \frac{1}{π} \cdot \int_{0}^{1} 1 - v^{2} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{π} (\int_{0}^{1} 1 d v - \int_{0}^{1} v^{2} d v)

\int_{0}^{1} 1 d v = 1

\int_{0}^{1} v^{2} d v = \frac{1}{3}

= \frac{1}{π} (1 - \frac{1}{3})

Simplify

\frac{1}{π} (1 - \frac{1}{3}) : \frac{2}{3 π}

= \frac{2}{3 π}

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What is the integral from 0 to 1/2 of cos^3(pix) ?
The integral from 0 to 1/2 of cos^3(pix) is 2/(3pi)