What is the integral of (4(ln(x))^2+x)/x ?

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

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integral of (4(ln(x))^2+x)/x

\int \frac{4 ( ln ( x ) ) ^{2} + x}{x} d x

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

Solution steps

\int \frac{4 ( ln ( x ) ) ^{2} + x}{x} d x

Simplify

= \int \frac{4 ln ^{2} ( x ) + x}{x} d x

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{4 ln ^{2} ( x )}{x} d x = \frac{4}{3} ln^{3} (x)

\int 1 d x = x

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

Add a constant to the solution

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

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