What is the integral of (3ln^3(x)+5ln(x^2))/x ?

The integral of (3ln^3(x)+5ln(x^2))/x is 3/4 ln^4(x)+5ln^2(x)+C

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integral of (3ln^3(x)+5ln(x^2))/x

\int \frac{3 ln ^{3} ( x ) + 5 ln ( x ^{2} )}{x} d x

\frac{3}{4} ln^{4} (x) + 5 ln^{2} (x) + C

Solution steps

\int \frac{3 ln ^{3} ( x ) + 5 ln ( x ^{2} )}{x} d x

Expand

\frac{3 ln ^{3} ( x ) + 5 ln ( x ^{2} )}{x} : \frac{3 ln ^{3} ( x )}{x} + \frac{5 ln ( x ^{2} )}{x}

= \int \frac{3 ln ^{3} ( x )}{x} + \frac{5 ln ( x ^{2} )}{x} 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{3 ln ^{3} ( x )}{x} d x + \int \frac{5 ln ( x ^{2} )}{x} d x

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

\int \frac{5 ln ( x ^{2} )}{x} d x = 5 ln^{2} (x)

= \frac{3}{4} ln^{4} (x) + 5 ln^{2} (x)

Add a constant to the solution

= \frac{3}{4} ln^{4} (x) + 5 ln^{2} (x) + C

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What is the integral of (3ln^3(x)+5ln(x^2))/x ?
The integral of (3ln^3(x)+5ln(x^2))/x is 3/4 ln^4(x)+5ln^2(x)+C