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

The integral of ((sin(x))/x)^2 is -1/x+(cos^2(x))/x+\Si(2x)+C

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

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

- \frac{1}{x} + \frac{cos ^{2} ( x )}{x} + Si (2 x) + C

Solution steps

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Rewrite using trig identities

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

Expand

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

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

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

\int \frac{1}{x ^{2}} d x = - \frac{1}{x}

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

= - \frac{1}{x} - (- \frac{cos ^{2} ( x )}{x} - Si (2 x))

Simplify

= - \frac{1}{x} + \frac{cos ^{2} ( x )}{x} + Si (2 x)

Add a constant to the solution

= - \frac{1}{x} + \frac{cos ^{2} ( x )}{x} + Si (2 x) + C

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What is the integral of ((sin(x))/x)^2 ?
The integral of ((sin(x))/x)^2 is -1/x+(cos^2(x))/x+\Si(2x)+C