What is the integral from 0 to pi of cos^4(2t) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to pi of cos^4(2t)

\int_{0}^{π} cos^{4} (2 t) d t

\frac{3 π}{8}

Decimal Notation

1.17809 \dots

Solution steps

\int_{0}^{π} cos^{4} (2 t) d t

Apply u-substitution

: \int_{0}^{2 π} cos^{4} (u) \frac{1}{2} d u

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

Take the constant out:

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

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

Simplify

cos^{4} (u) : cos^{3} (u) cos (u)

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

Apply Integration By Parts

: [cos^{3} (u) sin (u) - \int - 3 cos^{2} (u) sin^{2} (u) d u]_{0}^{2 π}

= \frac{1}{2} [cos^{3} (u) sin (u) - \int - 3 cos^{2} (u) sin^{2} (u) d u]_{0}^{2 π}

\int - 3 cos^{2} (u) sin^{2} (u) d u = - \frac{3}{8} (u - \frac{1}{4} sin (4 u))

= \frac{1}{2} [cos^{3} (u) sin (u) - (- \frac{3}{8} (u - \frac{1}{4} sin (4 u)))]_{0}^{2 π}

Simplify

\frac{1}{2} [cos^{3} (u) sin (u) - (- \frac{3}{8} (u - \frac{1}{4} sin (4 u)))]_{0}^{2 π} : \frac{1}{2} [\frac{1}{32} (32 cos^{3} (u) sin (u) + 3 (4 u - sin (4 u)))]_{0}^{2 π}

= \frac{1}{2} [\frac{1}{32} (32 cos^{3} (u) sin (u) + 3 (4 u - sin (4 u)))]_{0}^{2 π}

Compute the boundaries:

\frac{3 π}{4}

= \frac{1}{2} \cdot \frac{3 π}{4}

Simplify

= \frac{3 π}{8}

\frac{d}{d x} ((x^{3} - x)^{3})

x^{2} y^{^{'}} + 2 x y = sin (3 x)

x \to 0 - lim (\frac{x + 94}{x})

\frac{d}{d t} (\frac{2 t}{t ^{2} + 1})

e^{5 x} cos (8 x)

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