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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

= (ln (x) + 2)^{2} - (- \frac{ln ^{3} ( x )}{3} - ln^{2} (x))

Simplify

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

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

Add a constant to the solution

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

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