Question: Evaluate $\int^1_0 \int^\sqrt{1+x^2}_0 \frac{dxdy}{1 + x^2 + y^2}$
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Subject : Applied Mathematics 2

Topic : Double Integration

Difficulty: High

mumbai university • 30 views
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modified 3 days ago  • written 4 days ago by gravatar for Ankit Pandey Ankit Pandey ♦♦ 10
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Solution:

$\mathrm{I}=\int_{0}^{1} \int_{0}^{\sqrt{1+x^{2}}} \frac{d x d y}{1+x^{2}+y^{2}}$

$I=\int_{0}^{1} \frac{1}{\sqrt{1+x^{2}}}\left[\tan ^{-1} \frac{y} {\sqrt{1+x^{2}}}\right]_{0}^{\sqrt{1+x^{2}}} \mathrm{dx}$

$\therefore \mathrm{I}=\int_{0}^{1} \frac{\pi}{4} \frac{1}{\sqrt{1+x^{2}}} \mathrm{d} x$

$\therefore I=\frac{\pi}{4}\left[\log \left(x+\sqrt{1+x^{2}}\right)\right]_{0}^{1}$

$\therefore I=\frac{\pi}{4} \log (1+\sqrt{2})$

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written 3 days ago by gravatar for Ankit Pandey Ankit Pandey ♦♦ 10
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