What is the integral of 1/(x^2)sin(2/x+3) ?

The integral of 1/(x^2)sin(2/x+3) is 1/2 cos(3+2/x)+C

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integral of 1/(x^2)sin(2/x+3)

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

\frac{1}{2} cos (3 + \frac{2}{x}) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{sin ( 3 ) cos ( \frac{2}{x} ) + cos ( 3 ) sin ( \frac{2}{x} )}{x ^{2}} d x

Expand

\frac{sin ( 3 ) cos ( \frac{2}{x} ) + cos ( 3 ) sin ( \frac{2}{x} )}{x ^{2}} : \frac{sin ( 3 ) cos ( \frac{2}{x} )}{x ^{2}} + \frac{cos ( 3 ) sin ( \frac{2}{x} )}{x ^{2}}

= \int \frac{sin ( 3 ) cos ( \frac{2}{x} )}{x ^{2}} + \frac{cos ( 3 ) sin ( \frac{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{sin ( 3 ) cos ( \frac{2}{x} )}{x ^{2}} d x + \int \frac{cos ( 3 ) sin ( \frac{2}{x} )}{x ^{2}} d x

\int \frac{sin ( 3 ) cos ( \frac{2}{x} )}{x ^{2}} d x = - \frac{1}{2} sin (3) sin (\frac{2}{x})

\int \frac{cos ( 3 ) sin ( \frac{2}{x} )}{x ^{2}} d x = \frac{1}{2} cos (3) cos (\frac{2}{x})

= - \frac{1}{2} sin (3) sin (\frac{2}{x}) + \frac{1}{2} cos (3) cos (\frac{2}{x})

Simplify

- \frac{1}{2} sin (3) sin (\frac{2}{x}) + \frac{1}{2} cos (3) cos (\frac{2}{x}) : \frac{1}{2} cos (3 + \frac{2}{x})

= \frac{1}{2} cos (3 + \frac{2}{x})

Add a constant to the solution

= \frac{1}{2} cos (3 + \frac{2}{x}) + C

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What is the integral of 1/(x^2)sin(2/x+3) ?
The integral of 1/(x^2)sin(2/x+3) is 1/2 cos(3+2/x)+C