First slide

Introduction to integration

Question

$\int \frac{d x}{{(a^{2} + x^{2})}^{\frac{3}{2}}} =$

Easy

A

$\frac{x}{{(a^{2} + x^{2})}^{\frac{3}{2}}} + c$
B

$\frac{x}{a^{2} {(a^{2} + x^{2})}^{\frac{3}{2}}} + c$
C

$\frac{1}{a^{2} {(a^{2} + x^{2})}^{\frac{3}{2}}} + c$
D

None of these

Solution

$\begin{array}{l} P u t x = a \tan θ \Rightarrow d x = a s e c^{2} θ d θ \\ ∴ I = \int \frac{a s e c^{2} θ}{{(a^{2} + a^{2} \tan^{2} θ)}^{\frac{3}{2}}} d θ \\ = \int \frac{a s e c^{2} θ}{a^{3} {(s e c^{2} θ)}^{\frac{3}{2}}} d θ \\ = \frac{1}{a^{2}} \int \frac{d θ}{s e c θ} \\ = \frac{1}{a^{2}} \int \cos θ d θ \\ = \frac{1}{a^{2}} \sin θ + c \\ = \frac{x}{a^{2} {(a^{2} + x^{2})}^{\frac{3}{2}}} + c \end{array}$

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