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Integral Calculus> Help me with these three integration prob...

question mark

1. sin^-1√[x/(x+a)] dx

2. (sin^-1√x - cos^-1 √x)/ ??(sin^-1√x + cos^-1 √x) dx

3. tan^-1{[√(1+x) - √(1-x)]/[√(1+x) + √(1-x)]} dx

Rohit G , 14 Years ago

Grade 11

1 Answers

Jitender Singh

Ans:

1.

$I = \int sin^{-1}\sqrt{\frac{x}{x+a}}dx$

Integration by Parts

$I = xsin^{-1}\sqrt{\frac{x}{x+a}}-\int \frac{1}{2}\sqrt{\frac{ax}{(x+a)^{2}}}dx$

$t = \sqrt{x}$

$dt = \frac{1}{2\sqrt{x}}dx$

$I = xsin^{-1}\sqrt{\frac{x}{x+a}}-\sqrt{a}\int\frac{t^{2}}{a+t^{2}}dt$

$I = xsin^{-1}\sqrt{\frac{x}{x+a}}-\sqrt{a}t +atan^{-1}(\frac{t}{\sqrt{a}}) +constant$

$I = xsin^{-1}\sqrt{\frac{x}{x+a}}-\sqrt{a}\sqrt{x}+atan^{-1}(\frac{\sqrt{x}}{\sqrt{a}}) +constant$

2.

$I = \int (\frac{sin^{-1}\sqrt{x}-cos^{-1}\sqrt{x}}{sin^{-1}\sqrt{x}+cos^{-1}\sqrt{x}})dx$

Simply apply the substitution method here, we have

$I = \frac{\sqrt{x(1-x)}+(x-1)sin^{-1}\sqrt{x}-xcos^{-1}\sqrt{x}}{sin^{-1}\sqrt{x}+cos^{-1}\sqrt{x}} + constant$

3.

$I = \int tan^{-1}(\frac{\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}})dx$

Integration by parts

$I = xtan^{-1}(\frac{\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}})-\frac{1}{2}\int\frac{x}{\sqrt{1-x^{2}}}dx$

$t = 1-x^{2}$

$dt = -2x.dx$

$I = xtan^{-1}(\frac{\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}})-\frac{1}{4}\int\frac{1}{\sqrt{t}}dt$

$I = xtan^{-1}(\frac{\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}})-\frac{\sqrt{t}}{2}+constant$

$I = xtan^{-1}(\frac{\sqrt{x+1}-\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}})-\frac{\sqrt{1-x^{2}}}{2}+constant$

Thanks & Regards

Jitender Singh

IIT Delhi

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