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

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

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integral of (xarctan(x))/(sqrt(x^2+1))

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

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

Solution steps

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

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

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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