I∫_0^a x√(a^2+x^2)dx Put t=a^2+x^2 dtdx=2x dt2=xdx ∴, I=∫dt2√(t) I= 12.t^ 12+1 12+1+C I= 12.√(t) 12+C I=√(t)+C I=[√(x ...

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Quadratic Formula

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Question

Solve $\int_{0}^{a} \frac{x}{\sqrt{a^{2} + x^{2}}} d x$

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Solution

$I \int_{0}^{a} \frac{x}{\sqrt{a^{2} + x^{2}}} d x$ Put $t = a^{2} + x^{2} ⟹ \frac{d t}{d x} = 2 x ⟹ \frac{d t}{2} = x d x$ $∴, I = \int \frac{d t}{2 \sqrt{t}}$ $I = \frac{1}{2} . \frac{t^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1} + C$ $I = \frac{1}{2} . \frac{\sqrt{t}}{\frac{1}{2}} + C I = \sqrt{t} + C I = [\sqrt{x^{2} + a^{2}}]_{0}^{a} I = a \sqrt{2} - a I = a (\sqrt{2} - 1)$

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