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

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

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

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

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

Solution steps

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

Simplify

= \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

\int \frac{x + 3}{( x - 1 ) ^{2}} d x

f (x) = 9 - 4 x, at x = - 1

\frac{d}{d x} (\frac{9 x}{4})

f (x) = 9 - x^{2}, g (x) = - 8 x

\int x^{5} d x

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