What is the integral of (x*arctan(x))/(sqrt(1+x^2)) ?

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

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integral of (x*arctan(x))/(sqrt(1+x^2))

\int \frac{x \cdot arctan ( x )}{1 + x ^{2}} d x

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

Solution steps

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

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

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

Apply Integration By Parts

: arctan (x) 1 + x^{2} - \int \frac{1}{x ^{2} + 1} d x

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

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

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

Add a constant to the solution

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

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