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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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