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

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

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

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

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

Solution steps

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

Simplify

\frac{1}{x ^{3}} sin (\frac{1}{x}) : \frac{sin ( \frac{1}{x} )}{x ^{3}}

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

Apply u-substitution

= \int - u sin (u) d u

Take the constant out:

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

= - \int u sin (u) d u

Apply Integration By Parts

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

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

= - (- u cos (u) - (- sin (u)))

Substitute back

u = \frac{1}{x}

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

Simplify

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

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

Add a constant to the solution

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

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