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

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

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

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

\frac{3 π}{4}

Decimal Notation

2.35619 \dots

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

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

Apply the Sum Rule:

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

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

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

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

= \frac{1}{2} (\frac{3 π}{2} + 0)

Simplify

\frac{1}{2} (\frac{3 π}{2} + 0) : \frac{3 π}{4}

= \frac{3 π}{4}

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