What is the integral of ((8ln(x)-4)^6)/x ?

The integral of ((8ln(x)-4)^6)/x is -1/56 (-8ln(x)+4)^7+C

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integral of ((8ln(x)-4)^6)/x

\int \frac{( 8 ln ( x ) - 4 ) ^{6}}{x} d x

- \frac{1}{56} (- 8 ln (x) + 4)^{7} + C

Solution steps

\int \frac{( 8 ln ( x ) - 4 ) ^{6}}{x} d x

Factor

8 ln (x) - 4 : - (- 8 ln (x) + 4)

= \int \frac{( - ( - 8 ln ( x ) + 4 ) ) ^{6}}{x} d x

(- (- 8 ln (x) + 4))^{6} = (- 8 ln (x) + 4)^{6}

= \int \frac{( - 8 ln ( x ) + 4 ) ^{6}}{x} d x

Apply u-substitution

: \int - \frac{u ^{6}}{8} d u

= \int - \frac{u ^{6}}{8} d u

Take the constant out:

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

= - \frac{1}{8} \cdot \int u^{6} d u

Apply the Power Rule

: \frac{u ^{7}}{7}

= - \frac{1}{8} \cdot \frac{u ^{7}}{7}

Substitute back

u = - 8 ln (x) + 4

= - \frac{1}{8} \cdot \frac{( - 8 ln ( x ) + 4 ) ^{7}}{7}

Simplify

- \frac{1}{8} \cdot \frac{( - 8 ln ( x ) + 4 ) ^{7}}{7} : - \frac{1}{56} (- 8 ln (x) + 4)^{7}

= - \frac{1}{56} (- 8 ln (x) + 4)^{7}

Add a constant to the solution

= - \frac{1}{56} (- 8 ln (x) + 4)^{7} + C

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What is the integral of ((8ln(x)-4)^6)/x ?
The integral of ((8ln(x)-4)^6)/x is -1/56 (-8ln(x)+4)^7+C