∫ √((1-x)/(1+x)) dx is: (a) √(1-x²) + sin⁻¹x + c ( – Class 11 Mathematics

Question 109 of 129

\displaystyle\int \sqrt{\dfrac{1-x}{1+x}}\, dx

(a)

\sqrt{1-x^2} + \sin^{-1}x + c

(b)

\sin^{-1}x - \sqrt{1-x^2} + c

(c)

\log\left|x + \sqrt{1-x^2}\right| - \sqrt{1-x^2} + c

(d)

\sqrt{1-x^2} + \log\left|x + \sqrt{1-x^2}\right| + c

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Rationalise the integrand by multiplying by $\sqrt{1-x}/\sqrt{1-x}$ , then split and integrate term by term.

Multiply numerator and denominator inside the square root by $(1-x)$ :

$\sqrt{\dfrac{1-x}{1+x}}=\sqrt{\dfrac{(1-x)^2}{(1+x)(1-x)}}=\dfrac{1-x}{\sqrt{1-x^2}}=\dfrac{1}{\sqrt{1-x^2}}-\dfrac{x}{\sqrt{1-x^2}}.$

Now integrate each piece: …

Q. $\displaystyle\int \sqrt{\dfrac{1-x}{1+x}}\, dx$ is: