What is the integral from 0 to pi/2 of cos^3(21x) ?

The integral from 0 to pi/2 of cos^3(21x) is 2/63

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integral from 0 to pi/2 of cos^3(21x)

\int_{0}^{\frac{π}{2}} cos^{3} (21 x) d x

\frac{2}{63}

Decimal Notation

0.03174 \dots

Solution steps

\int_{0}^{\frac{π}{2}} cos^{3} (21 x) d x

Apply u-substitution

: \int_{0}^{\frac{21 π}{2}} cos^{3} (u) \frac{1}{21} d u

= \int_{0}^{\frac{21 π}{2}} cos^{3} (u) \frac{1}{21} d u

Take the constant out:

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

= \frac{1}{21} \cdot \int_{0}^{\frac{21 π}{2}} cos^{3} (u) d u

Rewrite using trig identities

= \frac{1}{21} \cdot \int_{0}^{\frac{21 π}{2}} (1 - sin^{2} (u)) cos (u) d u

Apply u-substitution

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

= \frac{1}{21} \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}{21} (\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}{21} (1 - \frac{1}{3})

Simplify

\frac{1}{21} (1 - \frac{1}{3}) : \frac{2}{63}

= \frac{2}{63}

t \to 0 lim (\frac{e ^{2 t} - 1}{t})

y = ln (30) + 3 0^{x + 4}

x \to 4 lim (x^{2} - 16)

\int 3 x - 8 d x

What is the integral from 0 to pi/2 of cos^3(21x) ?
The integral from 0 to pi/2 of cos^3(21x) is 2/63