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

The integral of (3x^5+x^2ln(x)) is (x^6)/2+1/3 x^3ln(x)-(x^3)/9+C

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

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

\frac{x ^{6}}{2} + \frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9} + C

Solution steps

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

\int 3 x^{5} d x = \frac{x ^{6}}{2}

\int x^{2} ln (x) d x = \frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}

= \frac{x ^{6}}{2} + \frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}

Add a constant to the solution

= \frac{x ^{6}}{2} + \frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9} + C

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What is the integral of (3x^5+x^2ln(x)) ?
The integral of (3x^5+x^2ln(x)) is (x^6)/2+1/3 x^3ln(x)-(x^3)/9+C