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$\int \left ( \frac{2a+x}{a+x} \right )\sqrt{\frac{a-x}{a+x}}dx=$

Option 1)
$\sqrt{a^{2}-x^{2}}-2a\sqrt{\frac{a-x}{a+x}}+c$

Option 2)
$-\sqrt{a^{2}-x^{2}}-2a\sqrt{\frac{a-x}{a+x}}+c$

Option 3)
$\frac{1}{a}\tan^{-1}\frac{x}{a}+ln|x+\sqrt{a^{2}-x^{2}}|+c$

Option 4)
$\frac{1}{2a}ln|\frac{a+x}{a-x}|+\sin^{-1}\frac{x}{a}+c$

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 $x=a\cos ^{2}\Theta +b\sin ^{2}\Theta$

Put $\theta = co{s^{ - 1}}\frac{x}{a}\left( { - a < x < a} \right)$ .Then $0 < \theta < \pi$

and $I = - a\int {\frac{{\left( {2 + \cos \theta } \right)\left( {1 - \cos \theta } \right)}}{{1 + \cos \theta }}} d\theta$

$=- a\int {\left\{ {\left( {1 - \cos \theta } \right) + \frac{{1 - \cos \theta }}{{1 + \cos \theta }}} \right\}} d\theta= - a\left( {2\tan \frac{\theta }{2} - \sin \theta } \right) + c$

$=\sqrt {{a^2} - {x^2}} - 2a\sqrt {\frac{{a - x}}{{a + x}}} + c$

Option 1)

$\sqrt{a^{2}-x^{2}}-2a\sqrt{\frac{a-x}{a+x}}+c$

Option 2)

$-\sqrt{a^{2}-x^{2}}-2a\sqrt{\frac{a-x}{a+x}}+c$

Option 3)

$\frac{1}{a}\tan^{-1}\frac{x}{a}+ln|x+\sqrt{a^{2}-x^{2}}|+c$

Option 4)

$\frac{1}{2a}ln|\frac{a+x}{a-x}|+\sin^{-1}\frac{x}{a}+c$

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gaurav

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