What is the integral of (cos(pi/(x^{31)}))/(x^{32)} ?

The integral of (cos(pi/(x^{31)}))/(x^{32)} is -1/(31pi)sin(pi/(x^{31)})+C

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integral of (cos(pi/(x^{31)}))/(x^{32)}

\int \frac{cos ( \frac{π}{x ^{31}} )}{x ^{32}} d x

- \frac{1}{31 π} sin (\frac{π}{x ^{31}}) + C

Solution steps

\int \frac{cos ( \frac{π}{x ^{31}} )}{x ^{32}} d x

Apply u-substitution

: \int \frac{cos ( \frac{π}{u} )}{31 u ^{2}} d u

= \int \frac{cos ( \frac{π}{u} )}{31 u ^{2}} d u

Take the constant out:

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

= \frac{1}{31} \cdot \int \frac{cos ( \frac{π}{u} )}{u ^{2}} d u

Apply u-substitution

: \int - \frac{cos ( v )}{π} d v

= \frac{1}{31} \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}{31} (- \frac{1}{π} \cdot \int cos (v) d v)

Use the common integral:

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

= \frac{1}{31} (- \frac{1}{π} sin (v))

Substitute back

= \frac{1}{31} (- \frac{1}{π} sin (\frac{π}{x ^{31}}))

Simplify

\frac{1}{31} (- \frac{1}{π} sin (\frac{π}{x ^{31}})) : - \frac{1}{31 π} sin (\frac{π}{x ^{31}})

= - \frac{1}{31 π} sin (\frac{π}{x ^{31}})

Add a constant to the solution

= - \frac{1}{31 π} sin (\frac{π}{x ^{31}}) + C

y = \frac{e ^{x}}{1 + e ^{x}}

\frac{d}{d x} (\frac{x ^{2} + 2 x - 23}{x - 4})

y = 9 cos (x)

\frac{x}{x + 2}

\frac{e ^{- s π}}{s ^{2} + 2 s + 2}

What is the integral of (cos(pi/(x^{31)}))/(x^{32)} ?
The integral of (cos(pi/(x^{31)}))/(x^{32)} is -1/(31pi)sin(pi/(x^{31)})+C