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

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

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integral of (sin(ln(x^6)))/x

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

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

Solution steps

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

Simplify

ln (x^{6}) : 6 ln (x)

= \int \frac{sin ( 6 ln ( x ) )}{x} d x

Apply u-substitution

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

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

Use the common integral:

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

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

Substitute back

u = 6 ln (x)

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

Simplify

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

Add a constant to the solution

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

\int \frac{1}{x ^{2} - 2 x - 3} d x

f (x) = - \frac{1}{4} x^{2}

f^{^{'}} (x) = \frac{x}{4}

\int 121 - t^{2} d t

f (x) = x^{2} - 5, at x = - 4

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