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

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

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/2 of 2cos^3(x)

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

\frac{4}{3}

Decimal

1.33333 \dots

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

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

Apply u-substitution

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

= 2 \cdot \int_{0}^{1} 1 - u^{2} d u

Apply the Sum Rule:

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

= 2 (\int_{0}^{1} 1 d u - \int_{0}^{1} u^{2} d u)

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

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

= 2 (1 - \frac{1}{3})

Simplify

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

= \frac{4}{3}

\int_{- 2}^{5} 6 x - x^{2} d x

\int_{1}^{t} \frac{ln ( x )}{x} d x

\int_{l}^{u} r \cdot e^{- t \cdot r} d t

\int_{4}^{5.2} ln (x) d x

\int_{2}^{3.8} ln (5 - x) d x

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