Consider the given integral. #10; #10; I=∫x√(a^2+x^2)dx #10; #10; #160; #10; #10;Let t=a^2+x^2 #10; #10; dtdx=0+2x #10; #1 ...

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Integration by Substitution

Solve ∫x√a2...

Question

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

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Solution

Consider the given integral.

$I = \int \frac{x}{\sqrt{a^{2} + x^{2}}} d x$

Let $t = a^{2} + x^{2}$

$\frac{d t}{d x} = 0 + 2 x$

$\frac{d t}{2} = x d x$

Therefore,

$I = \frac{1}{2} \int \frac{d t}{\sqrt{t}}$

$I = \frac{1}{2} \times 2 \sqrt{t} + C$

$I = \sqrt{t} + C$

On putting the value of $t$ , we get

$I = \sqrt{a^{2} + x^{2}} + C$

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