If $\int \frac{\sqrt{4 + x^{2}}}{x^{6}} d x$ = $\frac{A {(4 + x^{2})}^{3 / 2} (B x^{2} - 6)}{x^{5}} + C$ , then A is:

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If $\int \frac{\sqrt{4 + x^{2}}}{x^{6}} d x$ = $\frac{A {(4 + x^{2})}^{3 / 2} (B x^{2} - 6)}{x^{5}} + C$ , then A is:

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AAI ATC Junior Executive 21 Feb 2023 Shift 2 Official Paper

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-120
$- \frac{1}{120}$
120
$\frac{1}{120}$

Answer 1

Given:

$\int \frac{\sqrt{4 + x^{2}}}{x^{6}} d x$ = $\frac{A {(4 + x^{2})}^{3 / 2} (B x^{2} - 6)}{x^{5}} + C$

Concept:

Use concept of Integration using substitution ..

Calculation:

$\int \frac{\sqrt{4 + x^{2}}}{x^{6}} d x$

Substitute $x = \frac{1}{t} ⟹ d x = \frac{- 1}{t^{2}} d t$ then

$= \int \frac{\sqrt{4 + \frac{1}{t^{2}}}}{\frac{1}{t^{6}}} \cdot (- \frac{1}{t^{2}}) d t$

$= - \int t^{3} \sqrt{4 t^{2} + 1} d t$

$= - \int t^{2} \cdot t \sqrt{4 t^{2} + 1} d t$

Integrate using multiplication rule.

$= - [t^{2} \cdot \int t \sqrt{4 t^{2} + 1} d t - \int (\frac{d}{d t} (t^{2}) \cdot \int t \sqrt{4 t^{2} + 1} d t) d t]$

Substitute 4t² + 1 = u ⇒ t dt = 1/8 du then

$= - [t^{2} \cdot \frac{1}{8} \int \sqrt{u} d u - \int (\frac{d}{d t} (t^{2}) \cdot \frac{1}{8} \int \sqrt{u} d u) d t]$

$= - [t^{2} \cdot \frac{1}{8} \frac{u^{3 / 2}}{3 / 2} - \int (2 t \cdot \frac{1}{8} \frac{u^{3 / 2}}{3 / 2}) d t]$

Undo substitution of u .

$= - [t^{2} \cdot \frac{1}{8} \frac{{(4 t^{2} + 1)}^{3 / 2}}{3 / 2} - \int (2 t \cdot \frac{1}{8} \frac{{(4 t^{2} + 1)}^{3 / 2}}{3 / 2}) d t]$

$= - [\frac{1}{12} t^{2} (4 t^{2} + 1)^{3 / 2} - \frac{1}{6} \int t (4 t^{2} + 1)^{3 / 2} d t]$

Again Substitute 4t2 + 1 = u ⇒ t dt = 1/8 du then

$= - [\frac{1}{12} t^{2} (4 t^{2} + 1)^{3 / 2} - \frac{1}{6} \int \frac{1}{8} (u)^{3 / 2} d u]$

$= - [\frac{1}{12} t^{2} (4 t^{2} + 1)^{3 / 2} - \frac{1}{48} \frac{(u)^{5 / 2}}{5 / 2}] + C$

Undo substitution of u .

$= - [\frac{1}{12} t^{2} (4 t^{2} + 1)^{3 / 2} - \frac{1}{120} (4 t^{2} + 1)^{5 / 2}] + C$

Undo substitution of t .

$= - [\frac{1}{12} \frac{1}{x^{2}} (\frac{4}{x^{2}} + 1)^{3 / 2} - \frac{1}{120} (\frac{4}{x^{2}} + 1)^{5 / 2}] + C$

$= - [\frac{1}{12} \frac{(4 + x^{2})^{3 / 2}}{x^{5}} - \frac{1}{120} \frac{(4 + x^{2})^{5 / 2}}{x^{5}}] + C$

$= \frac{\frac{1}{120} {(4 + x^{2})}^{3 / 2} (x^{2} - 6)}{x^{5}} + C$

Hence A = $\frac{1}{120}$ .

Hence the option (4) is correct.

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