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

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

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integral of x*arctan(sqrt(x^2-1))

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

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

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{x}{2 x ^{2} - 1} d x = \frac{1}{2} x^{2} - 1

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

Add a constant to the solution

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

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