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Integration by Substitution

∫x/x2 + a2 dx

Question

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

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Solution

$\int \frac{x}{x^{2} + a^{2}} d x = \frac{1}{2} \int \frac{2 x}{x^{2} + a^{2}} d x = \frac{1}{2} ln | x^{2} + a^{2} | + c$

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Similar questions

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

\int \frac{1}{(x \sqrt{x^{2} + a^{2}})} d x

\int \frac{1}{\sqrt{x^{2} + a^{2}}} d x = l o g | x + \sqrt{x^{2} + a^{2}} | + c

\int \sqrt{x^{2} - a^{2}} d x = \frac{x}{2} \sqrt{x^{2} - a^{2}} - \frac{a^{2}}{2} log | x + \sqrt{x^{2} - a^{2}} | + c

Q. Prove that :-

\sqrt{x^{2} + a^{2}} d x = \frac{x}{2} \sqrt{x^{2} + a^{2}} + \frac{a^{2}}{2} log ∣ ∣ x + \sqrt{x^{2} + a^{2}} ∣ ∣ + c

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