What is the integral of 1/(x^2)sin(2/x)cos(2/x) ?

The integral of 1/(x^2)sin(2/x)cos(2/x) is 1/4 cos^2(2/x)+C

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integral of 1/(x^2)sin(2/x)cos(2/x)

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

\frac{1}{4} cos^{2} (\frac{2}{x}) + C

Solution steps

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

Apply u-substitution

: \int \frac{u}{2} d u

= \int \frac{u}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= \frac{1}{2} \cdot \frac{u ^{2}}{2}

Substitute back

u = cos (\frac{2}{x})

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

Simplify

\frac{1}{2} \cdot \frac{cos ^{2} ( \frac{2}{x} )}{2} : \frac{1}{4} cos^{2} (\frac{2}{x})

= \frac{1}{4} cos^{2} (\frac{2}{x})

Add a constant to the solution

= \frac{1}{4} cos^{2} (\frac{2}{x}) + C

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What is the integral of 1/(x^2)sin(2/x)cos(2/x) ?
The integral of 1/(x^2)sin(2/x)cos(2/x) is 1/4 cos^2(2/x)+C