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

The integral of (sin^2(x))/x is 1/2 (ln|x|-\Ci(2x))+C

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integral of (sin^2(x))/x

integral

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

\frac{1}{2} (ln ∣ x ∣ - Ci (2 x)) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{1 - cos ( 2 x )}{2 x} 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 \frac{1 - cos ( 2 x )}{x} d x

Expand

\frac{1 - cos ( 2 x )}{x} : \frac{1}{x} - \frac{cos ( 2 x )}{x}

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{cos ( 2 x )}{x} d x = Ci (2 x)

= \frac{1}{2} (ln ∣ x ∣ - Ci (2 x))

Add a constant to the solution

= \frac{1}{2} (ln ∣ x ∣ - Ci (2 x)) + C

What is the integral of (sin^2(x))/x ?
The integral of (sin^2(x))/x is 1/2 (ln|x|-\Ci(2x))+C