Evaluate: {x²}{(x²+2)(x²+3)} \, dx

Correct Answer is : −2tan⁡−1(x2)+3tan⁡−1(x3)+C-{2} ^{-1}≤ft({x}{{2}}) + {3} ^{-1}≤ft({x}{{3}}) + C−2tan−1(2x)+3tan−1(3x)+C

{2} ^{-1}≤ft({x}{{2}}) + {3} ^{-1}≤ft({x}{{3}}) + C

-{2} ^{-1}≤ft({x}{{2}}) - {3} ^{-1}≤ft({x}{{3}}) + C

-{2} ^{-1}≤ft({x}{{2}}) + {3} ^{-1}≤ft({x}{{3}}) + C

Correct Answer is : −2tan⁡−1(x2)+3tan⁡−1(x3)+C-{2} ^{-1}≤ft({x}{{2}}) + {3} ^{-1}≤ft({x}{{3}}) + C−2tan−1(2x)+3tan−1(3x)+C

Test Index

UPSC NDA Math Model Paper 8

Question : 91 of 120

Marks: +1, -0

Evaluate:

\int \frac{x^{2}}{(x^{2}+2)(x^{2}+3)} \, dx

$\sqrt{2} \tan^{-1}\left(\frac{x}{\sqrt{2}}\right) + \sqrt{3} \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) + C$
$-\sqrt{2} \tan^{-1}\left(\frac{x}{\sqrt{2}}\right) - \sqrt{3} \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) + C$
$-\sqrt{2} \tan^{-1}\left(\frac{x}{\sqrt{2}}\right) + \sqrt{3} \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) + C$
None of these

Solution:

Here we have to find the value of

\int \frac{x^{2}}{(x^{2}+2)(x^{2}+3)} \, dx

Let

x^{2}=t

\Rightarrow \frac{x^{2}}{(x^{2}+2)(x^{2}+3)} = \frac{t}{(t+2)(t+3)}

Let

\frac{t}{(t+2)(t+3)} = \frac{A}{t+2} + \frac{B}{t+3}

⇒ t = A (t + 3) + B (t + 2)

By putting t = - 2 on both the sides of (1) we get A = - 2

By putting t = - 3 on both the sides of (1) we get B = 3

\frac{t}{(t+2)(t+3)} = \frac{-2}{t+2} + \frac{3}{t+3}

By substituting

x^{2}=t

in the above equation we get

\Rightarrow \frac{x^{2}}{(x^{2}+2)(x^{2}+3)} = \frac{-2}{x^{2}+2} + \frac{3}{x^{2}+3}

\Rightarrow \int \frac{x^{2}}{(x^{2}+2)(x^{2}+3)} \, dx

= -2 \int \frac{dx}{x^{2}+2} + 3 \int \frac{dx}{x^{2}+3}

As we know that,

\int \frac{dx}{x^{2}+a^{2}} = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C

where

C

is a constant

\Rightarrow \int \frac{x^{2}}{(x^{2}+2)(x^{2}+3)} \, dx

= \frac{-2}{\sqrt{2}} \tan^{-1}\left(\frac{x}{\sqrt{2}}\right)

+ \frac{3}{\sqrt{3}} \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) + C

= -\sqrt{2} \tan^{-1}\left(\frac{x}{\sqrt{2}}\right) + \sqrt{3} \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) + C

Where

C

is a constant

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