Mathematics

Solve $$\displaystyle\int \dfrac{dx}{(x^2 + a^2)^{3/2}}$$


SOLUTION
$$\int \dfrac{dx}{(x^2 + a^2)^{3/2}}$$
Let x = a tan y.
$$dx = a sec^2 y dy$$
$$\int \dfrac{a \, sec^2 y \, dy}{(a^2 tan^2 y + a^2)^{3/2}}$$
$$= \int \dfrac{a \, sec^2 y}{(a^2 sec^2 y)^{3/2}}$$
$$= \int \dfrac{a \, sec^2 y}{a^3 sec^3 y}$$
$$= \dfrac{1}{a^2} \int cos y$$
$$\dfrac{1}{a^2} sin y + c$$
$$\dfrac{x}{a} = tan y$$
$$cot y = \dfrac{a}{x}$$
$$cosec y = \sqrt{1 + \dfrac{a^2}{x^2}}$$
$$= \sqrt{\dfrac{x^2 = a^2}{x}} = \dfrac{1}{a^2} \dfrac{x}{\sqrt{x^2 + a^2}} + c$$
$$sin y = \dfrac{x}{\sqrt{x^2 + a^2}}$$
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Subjective Medium Published on 17th 09, 2020
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