What is the integral of xarcsin(1/x) ?

The integral of xarcsin(1/x) is 1/2 x^2arcsin(1/x)+1/2 sqrt(x^2-1)+C

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integral of xarcsin(1/x)

\int x arcsin (\frac{1}{x}) d x

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

Solution steps

\int x arcsin (\frac{1}{x}) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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