What is the integral of (sin(1/u))/(u^3) ?

The integral of (sin(1/u))/(u^3) is (cos(1/u))/u-sin(1/u)+C

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integral of (sin(1/u))/(u^3)

\int \frac{sin ( \frac{1}{u} )}{u ^{3}} d u

\frac{cos ( \frac{1}{u} )}{u} - sin (\frac{1}{u}) + C

Solution steps

\int \frac{sin ( \frac{1}{u} )}{u ^{3}} d u

Apply u-substitution

= \int - v sin (v) d v

Take the constant out:

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

= - \int v sin (v) d v

Apply Integration By Parts

= - (- v cos (v) - \int - cos (v) d v)

\int - cos (v) d v = - sin (v)

= - (- v cos (v) - (- sin (v)))

Substitute back

v = \frac{1}{u}

= - (- \frac{1}{u} cos (\frac{1}{u}) - (- sin (\frac{1}{u})))

Simplify

- (- \frac{1}{u} cos (\frac{1}{u}) - (- sin (\frac{1}{u}))) : \frac{cos ( \frac{1}{u} )}{u} - sin (\frac{1}{u})

= \frac{cos ( \frac{1}{u} )}{u} - sin (\frac{1}{u})

Add a constant to the solution

= \frac{cos ( \frac{1}{u} )}{u} - sin (\frac{1}{u}) + C

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