What is the integral of (sin(1/(x^6)))/(x^7) ?

The integral of (sin(1/(x^6)))/(x^7) is 1/6 cos(1/(x^6))+C

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integral of (sin(1/(x^6)))/(x^7)

\int \frac{sin ( \frac{1}{x ^{6}} )}{x ^{7}} d x

\frac{1}{6} cos (\frac{1}{x ^{6}}) + C

Solution steps

\int \frac{sin ( \frac{1}{x ^{6}} )}{x ^{7}} d x

Apply u-substitution

: \int \frac{sin ( \frac{1}{u} )}{6 u ^{2}} d u

= \int \frac{sin ( \frac{1}{u} )}{6 u ^{2}} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{sin ( \frac{1}{u} )}{u ^{2}} d u

Apply u-substitution

: \int - sin (v) d v

= \frac{1}{6} \cdot \int - sin (v) d v

Take the constant out:

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

= \frac{1}{6} (- \int sin (v) d v)

Use the common integral:

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

= \frac{1}{6} (- (- cos (v)))

Substitute back

= \frac{1}{6} (- (- cos (\frac{1}{x ^{6}})))

Simplify

= \frac{1}{6} cos (\frac{1}{x ^{6}})

Add a constant to the solution

= \frac{1}{6} cos (\frac{1}{x ^{6}}) + C

\frac{d}{d x} (\frac{1}{2} cos^{2} (x))

f (x) = \frac{3}{x ^{\frac{1}{2}}}

x \to 3 + lim (\frac{x + 2}{x + 3})

(- 6, - 7), (- 24, - 7)

\int e^{3 x} e^{- 5 x} d x

What is the integral of (sin(1/(x^6)))/(x^7) ?
The integral of (sin(1/(x^6)))/(x^7) is 1/6 cos(1/(x^6))+C