What is the integral of (cos(pi/(x^5)))/(x^6) ?

The integral of (cos(pi/(x^5)))/(x^6) is -1/(5pi)sin(pi/(x^5))+C

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integral of (cos(pi/(x^5)))/(x^6)

\int \frac{cos ( \frac{π}{x ^{5}} )}{x ^{6}} d x

- \frac{1}{5 π} sin (\frac{π}{x ^{5}}) + C

Solution steps

\int \frac{cos ( \frac{π}{x ^{5}} )}{x ^{6}} d x

Apply u-substitution

: \int \frac{cos ( \frac{π}{u} )}{5 u ^{2}} d u

= \int \frac{cos ( \frac{π}{u} )}{5 u ^{2}} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{cos ( \frac{π}{u} )}{u ^{2}} d u

Apply u-substitution

: \int - \frac{cos ( v )}{π} d v

= \frac{1}{5} \cdot \int - \frac{cos ( v )}{π} d v

Take the constant out:

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

= \frac{1}{5} (- \frac{1}{π} \cdot \int cos (v) d v)

Use the common integral:

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

= \frac{1}{5} (- \frac{1}{π} sin (v))

Substitute back

= \frac{1}{5} (- \frac{1}{π} sin (\frac{π}{x ^{5}}))

Simplify

\frac{1}{5} (- \frac{1}{π} sin (\frac{π}{x ^{5}})) : - \frac{1}{5 π} sin (\frac{π}{x ^{5}})

= - \frac{1}{5 π} sin (\frac{π}{x ^{5}})

Add a constant to the solution

= - \frac{1}{5 π} sin (\frac{π}{x ^{5}}) + C

\int sin (ln (x)) d x

\int 0.5 x d x

y^{^{''}} + 9 y = cos (3 x)

x \to - 8 lim (x (8))

\frac{d}{d x} (\frac{1}{x} - ln (x))

What is the integral of (cos(pi/(x^5)))/(x^6) ?
The integral of (cos(pi/(x^5)))/(x^6) is -1/(5pi)sin(pi/(x^5))+C