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

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

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

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

\frac{4}{3}

Decimal

1.33333 \dots

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

= \int_{- 1}^{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

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

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

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

= 2 - \frac{2}{3}

Simplify

2 - \frac{2}{3} : \frac{4}{3}

= \frac{4}{3}

\int_{0}^{\frac{π}{2}} cos (θ) sin (θ) d θ

\int_{- 1}^{1} x cos (x) d x

\int_{- π}^{π} x cos (n x) d x

\int_{0}^{π} x^{2} d x

\int_{0}^{π} cos^{2} (θ) sin (θ) d θ

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