What is the integral of 2sin^2(x/2) ?

The integral of 2sin^2(x/2) is x-sin(x)+C

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integral of 2sin^2(x/2)

\int 2 sin^{2} (\frac{x}{2}) d x

x - sin (x) + C

Solution steps

\int 2 sin^{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 sin^{2} (\frac{x}{2}) d x

Rewrite using trig identities

= 2 \cdot \int \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 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 1 d x - \int cos (x) d x)

\int 1 d x = x

\int cos (x) d x = sin (x)

= 2 \cdot \frac{1}{2} (x - sin (x))

Simplify

2 \cdot \frac{1}{2} (x - sin (x)) : x - sin (x)

= x - sin (x)

Add a constant to the solution

= x - sin (x) + C

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