What is the integral from 0 to pi/8 of 3cos^5(4x) ?

The integral from 0 to pi/8 of 3cos^5(4x) is 2/5

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integral from 0 to pi/8 of 3cos^5(4x)

\int_{0}^{\frac{π}{8}} 3 cos^{5} (4 x) d x

\frac{2}{5}

Decimal

0.4

Solution steps

\int_{0}^{\frac{π}{8}} 3 cos^{5} (4 x) d x

Take the constant out:

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

= 3 \cdot \int_{0}^{\frac{π}{8}} cos^{5} (4 x) d x

Apply u-substitution

= 3 \cdot \int_{0}^{\frac{π}{2}} cos^{5} (u) \frac{1}{4} d u

Take the constant out:

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

= 3 \cdot \frac{1}{4} \cdot \int_{0}^{\frac{π}{2}} cos^{5} (u) d u

Rewrite using trig identities

= 3 \cdot \frac{1}{4} \cdot \int_{0}^{\frac{π}{2}} (1 - sin^{2} (u))^{2} cos (u) d u

Apply u-substitution

= 3 \cdot \frac{1}{4} \cdot \int_{0}^{1} (1 - v^{2})^{2} d v

Expand

(1 - v^{2})^{2} : 1 - 2 v^{2} + v^{4}

= 3 \cdot \frac{1}{4} \cdot \int_{0}^{1} 1 - 2 v^{2} + v^{4} 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

= 3 \cdot \frac{1}{4} (\int_{0}^{1} 1 d v - \int_{0}^{1} 2 v^{2} d v + \int_{0}^{1} v^{4} d v)

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

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

\int_{0}^{1} v^{4} d v = \frac{1}{5}

= 3 \cdot \frac{1}{4} (1 - \frac{2}{3} + \frac{1}{5})

Simplify

3 \cdot \frac{1}{4} (1 - \frac{2}{3} + \frac{1}{5}) : \frac{2}{5}

= \frac{2}{5}

n = 0 \sum \infty \frac{x ^{2 n}}{2 n !}

\int 2 + 60 x - 5 x^{2} d x

in v erse l a pl a ce e^{- 5 s} \frac{1}{1 + s}

x y^{^{'}} - y = \frac{1}{2} x ln (x)

\int \frac{e ^{a x} + b}{e ^{a x} - b} d x

What is the integral from 0 to pi/8 of 3cos^5(4x) ?
The integral from 0 to pi/8 of 3cos^5(4x) is 2/5