What is the integral of x^2ln(x^2+1) ?

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

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integral of x^2ln(x^2+1)

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

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

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{2 x ^{4}}{3 ( x ^{2} + 1 )} d x = \frac{2}{3} (\frac{x ^{3}}{3} - x + arctan (x))

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

Add a constant to the solution

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

e^{- x} x^{2} + 3 e^{- x} - e^{- x}

x y^{^{'}} + y + x = 0, y (1) = 1

\frac{\partial}{\partial x} (ln (x^{5} - y^{4}))

\frac{d}{d x} (a cos (3 x))

(3 x^{2})^{2 x}

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