What is the integral of 1/(7x(ln(5x))^6) ?

The integral of 1/(7x(ln(5x))^6) is -1/(35ln^5(5x))+C

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integral of 1/(7x(ln(5x))^6)

\int \frac{1}{7 x ( ln ( 5 x ) ) ^{6}} d x

- \frac{1}{35 ln ^{5} ( 5 x )} + C

Solution steps

\int \frac{1}{7 x ( ln ( 5 x ) ) ^{6}} d x

Take the constant out:

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

= \frac{1}{7} \cdot \int \frac{1}{x ( ln ( 5 x ) ) ^{6}} d x

Simplify

= \frac{1}{7} \cdot \int \frac{1}{x ln ^{6} ( 5 x )} d x

Apply u-substitution

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

= \frac{1}{7} \cdot \int \frac{1}{u ln ^{6} ( u )} d u

Apply u-substitution

: \int \frac{1}{v ^{6}} d v

= \frac{1}{7} \cdot \int \frac{1}{v ^{6}} d v

Apply the Power Rule

: - \frac{1}{5 v ^{5}}

= \frac{1}{7} (- \frac{1}{5 v ^{5}})

Substitute back

= \frac{1}{7} (- \frac{1}{5 ln ^{5} ( 5 x )})

Simplify

\frac{1}{7} (- \frac{1}{5 ln ^{5} ( 5 x )}) : - \frac{1}{35 ln ^{5} ( 5 x )}

= - \frac{1}{35 ln ^{5} ( 5 x )}

Add a constant to the solution

= - \frac{1}{35 ln ^{5} ( 5 x )} + C

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What is the integral of 1/(7x(ln(5x))^6) ?
The integral of 1/(7x(ln(5x))^6) is -1/(35ln^5(5x))+C