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

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

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

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

4

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

= 3 \cdot \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

= 3 \cdot \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

= 3 (\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}

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

Simplify

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

= 4

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