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Evaluate \int ln\left ( \sqrt{1-x} +\sqrt{1+x}\right )dx.

  • Option 1)

    xln\left ( \sqrt{1-x}+\sqrt{1+x} \right )+\frac{x}{2}+\frac{1}{2}\sin^{-1}x+c

  • Option 2)

    xln\left ( \sqrt{1-x}+\sqrt{1+x} \right )-\frac{x}{2}+\frac{1}{2}\sin^{-1}x+c

  • Option 3)

    xln\left ( \sqrt{1-x}+\sqrt{1+x} \right )-\frac{x}{2}-\frac{1}{2}\sin^{-1}x+c

  • Option 4)

    -xln\left ( \sqrt{1-x}+\sqrt{1+x} \right )-\frac{x}{2}+\frac{1}{2}\sin^{-1}x+c

 

Answers (1)

best_answer

As we learnt

Special type of indefinite integration -

Integrals of the form :

(i)f(\sqrt{a^{2}-x^{2}}) , (ii)f(\sqrt{x^{2}-a^{2}})

(iii)f(\sqrt{a^{2}+x^{2}}), (iv)f(a^{2}+x^{2})

(v)f\left ( \sqrt{\frac{a-x}{a+x}}\right ), (vi)f\left ( \sqrt{\frac{a+x}{a-x}}\right )

(vii)f\left ( \sqrt{\frac{x-a}{b-x}} \right ), (viii)f\left ( \sqrt{(x-a)(x-b)}\right )

- wherein

Working rule :

for (i) put x=a\sin \Theta or a \cos \Theta

for (ii) Put x=a\sec \Theta or a \, cosec\Theta

for (iii) and (iv) Put x=a\tan \Theta or a\cot \Theta

for (v) and (vi) Put x=a\cos 2 \Theta

for (vii) and (viii) Put x=a\cos ^{2}\Theta +b\sin ^{2}\Theta

 

 

 If we take u=\ln \left( {\sqrt {1 - x} + \sqrt {1 + x} } \right)as the first function and v = 1 as the second function then

\ln \left( {\sqrt {1 - x} + \sqrt {1 + x} } \right)\int {1dx}-\int {\left( {\frac{d}{{dx}}\left( {\ln \left( {\sqrt {1 - x} + \sqrt {1 + x} } \right)} \right)\int {1dx} } \right)dx}

=x\ln \left( {\sqrt {1 - x} + \sqrt {1 + x} } \right)-\int {\frac{1}{{\left( {\sqrt {1 - x} + \sqrt {1 + x} } \right)}}} \left( { - \frac{1}{{2\sqrt {1 - x} }} + \frac{1}{{2\sqrt {1 + x} }}} \right)xdx=x\ln \left( {\sqrt {1 - x} + \sqrt {1 + x} } \right)-\frac{1}{2}\int {x\frac{{\sqrt {1 - {x^2}} - 1}}{{x\sqrt {1 - {x^2}} }}dx}

=x\ln \left( {\sqrt {1 - x} + \sqrt {1 + x} } \right)-\frac{1}{2}\int {dx} + \frac{1}{2}\int {\frac{1}{{\sqrt {1 - {x^2}} }}dx}

=x\ln \left( {\sqrt {1 - x} + \sqrt {1 + x} } \right)-\frac{x}{2} + \frac{1}{2}{\sin ^{ - 1}}x + c


Option 1)

xln\left ( \sqrt{1-x}+\sqrt{1+x} \right )+\frac{x}{2}+\frac{1}{2}\sin^{-1}x+c

Option 2)

xln\left ( \sqrt{1-x}+\sqrt{1+x} \right )-\frac{x}{2}+\frac{1}{2}\sin^{-1}x+c

Option 3)

xln\left ( \sqrt{1-x}+\sqrt{1+x} \right )-\frac{x}{2}-\frac{1}{2}\sin^{-1}x+c

Option 4)

-xln\left ( \sqrt{1-x}+\sqrt{1+x} \right )-\frac{x}{2}+\frac{1}{2}\sin^{-1}x+c

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