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

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

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

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

π

Decimal Notation

3.14159 \dots

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

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

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int_{0}^{π} 1 + cos (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

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

\int_{0}^{π} 1 d x = π

\int_{0}^{π} cos (x) d x = 0

= 2 \cdot \frac{1}{2} (π + 0)

Simplify

2 \cdot \frac{1}{2} (π + 0) : π

= π

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y^{^{''}} + 5 y^{^{'}} + 11 y = 33

\frac{d y}{d t} = 3 cos (t) - 5 y

\frac{\partial}{\partial x} (sin (12 x) cos (2 y))

\int_{6}^{8} \frac{74}{( x - 6 ) ^{3}} d x

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