What is the integral of (sin((-5)/x))/(11x^2) ?

The integral of (sin((-5)/x))/(11x^2) is -1/55 cos(5/x)+C

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integral of (sin((-5)/x))/(11x^2)

\int \frac{sin ( \frac{- 5}{x} )}{11 x ^{2}} d x

- \frac{1}{55} cos (\frac{5}{x}) + C

Solution steps

\int \frac{sin ( \frac{- 5}{x} )}{11 x ^{2}} d x

Take the constant out:

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

= \frac{1}{11} \cdot \int \frac{sin ( \frac{- 5}{x} )}{x ^{2}} d x

Simplify

\frac{sin ( \frac{- 5}{x} )}{x ^{2}} : - \frac{sin ( \frac{5}{x} )}{x ^{2}}

= \frac{1}{11} \cdot \int - \frac{sin ( \frac{5}{x} )}{x ^{2}} d x

Take the constant out:

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

= \frac{1}{11} (- \int \frac{sin ( \frac{5}{x} )}{x ^{2}} d x)

Apply u-substitution

: \int - \frac{sin ( u )}{5} d u

= \frac{1}{11} (- \int - \frac{sin ( u )}{5} d u)

Take the constant out:

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

= \frac{1}{11} (- (- \frac{1}{5} \cdot \int sin (u) d u))

Use the common integral:

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

= \frac{1}{11} (- (- \frac{1}{5} (- cos (u))))

Substitute back

u = \frac{5}{x}

= \frac{1}{11} (- (- \frac{1}{5} (- cos (\frac{5}{x}))))

Simplify

\frac{1}{11} (- (- \frac{1}{5} (- cos (\frac{5}{x})))) : - \frac{1}{55} cos (\frac{5}{x})

= - \frac{1}{55} cos (\frac{5}{x})

Add a constant to the solution

= - \frac{1}{55} cos (\frac{5}{x}) + C

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What is the integral of (sin((-5)/x))/(11x^2) ?
The integral of (sin((-5)/x))/(11x^2) is -1/55 cos(5/x)+C