What is the integral of ((2ln(x)+3)^3)/x ?

The integral of ((2ln(x)+3)^3)/x is 1/8 (2ln(x)+3)^4+C

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integral of ((2ln(x)+3)^3)/x

\int \frac{( 2 ln ( x ) + 3 ) ^{3}}{x} d x

\frac{1}{8} (2 ln (x) + 3)^{4} + C

Solution steps

\int \frac{( 2 ln ( x ) + 3 ) ^{3}}{x} d x

Apply u-substitution

: \int \frac{u ^{3}}{2} d u

= \int \frac{u ^{3}}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= \frac{1}{2} \cdot \frac{u ^{4}}{4}

Substitute back

u = 2 ln (x) + 3

= \frac{1}{2} \cdot \frac{( 2 ln ( x ) + 3 ) ^{4}}{4}

Simplify

\frac{1}{2} \cdot \frac{( 2 ln ( x ) + 3 ) ^{4}}{4} : \frac{1}{8} (2 ln (x) + 3)^{4}

= \frac{1}{8} (2 ln (x) + 3)^{4}

Add a constant to the solution

= \frac{1}{8} (2 ln (x) + 3)^{4} + C

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What is the integral of ((2ln(x)+3)^3)/x ?
The integral of ((2ln(x)+3)^3)/x is 1/8 (2ln(x)+3)^4+C