What is the integral of x/((x^2-1)^{1/2)} ?

The integral of x/((x^2-1)^{1/2)} is (x^2)/((x^2-1)^{1/2)}-1/((x^2-1)^{1/2)}+C

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integral of x/((x^2-1)^{1/2)}

integral

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Add a constant to the solution

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

What is the integral of x/((x^2-1)^{1/2)} ?
The integral of x/((x^2-1)^{1/2)} is (x^2)/((x^2-1)^{1/2)}-1/((x^2-1)^{1/2)}+C