The value of integral {{x²+1}≤ft[ ≤ft(x²+1)-2 x]}{x^{4}} d x is equal to...

Correct Answer is : (1+1x2)3/2[−13log⁡(1+1x2)+29]+c≤ft(1+{1}{x²})³ / 2}≤ft[-1/3 ≤ft(1+{1}{x²})+2/9]+c(1+x21)3/2[3−1log(1+x21)+92]+c

≤ft(1+{1}{x²})³ / 2}≤ft[-1/3 ≤ft(1+{1}{x²})+2/9]+c

≤ft(1+{1}{x²})³ / 2}≤ft[-1/3 ≤ft(1+{1}{x²})-2/9]+c

≤ft(1+{1}{x²})³ / 2}≤ft[-1/3 ≤ft(1+{1}{x²})+2/3]+c

≤ft(1+{1}{x²})³ / 2}≤ft[-1/3 ≤ft(1+{1}{x²})-2/3]+c

Correct Answer is : (1+1x2)3/2[−13log⁡(1+1x2)+29]+c≤ft(1+{1}{x²})³ / 2}≤ft[-1/3 ≤ft(1+{1}{x²})+2/9]+c(1+x21)3/2[3−1log(1+x21)+92]+c

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MHTCET 19 Apr 2026 Shift 1 Paper

Question : 129 of 150

Marks: +1, -0

The value of integral

\int \frac{\sqrt{x^{2}+1}\left[\log \left(x^{2}+1\right)-2 \log x\right]}{x^{4}} d x

is equal to...

$\left(1+\frac{1}{x^{2}}\right)^{3 / 2}\left[\frac{-1}{3} \log \left(1+\frac{1}{x^{2}}\right)+\frac{2}{9}\right]+c$
$\left(1+\frac{1}{x^{2}}\right)^{3 / 2}\left[\frac{-1}{3} \log \left(1+\frac{1}{x^{2}}\right)-\frac{2}{9}\right]+c$
$\left(1+\frac{1}{x^{2}}\right)^{3 / 2}\left[\frac{-1}{3} \log \left(1+\frac{1}{x^{2}}\right)+\frac{2}{3}\right]+c$
$\left(1+\frac{1}{x^{2}}\right)^{3 / 2}\left[\frac{-1}{3} \log \left(1+\frac{1}{x^{2}}\right)-\frac{2}{3}\right]+c$

Solution:

Concept:

Use logarithm properties to simplify the integrand, then apply substitution and integration by parts.

Explanation:

First, combine the logarithmic terms:

\log\left(x^{2}+1\right)-2\log x = \log\left(\frac{x^{2}+1}{x^{2}}\right) = \log\left(1+\frac{1}{x^{2}}\right).

Since

\log x

is defined,

x>0

. Hence,

\frac{\sqrt{x^{2}+1}}{x^{4}} = \frac{\sqrt{1+\frac{1}{x^{2}}}}{x^{3}}.

Let

t = 1+\frac{1}{x^{2}}, \quad dt = -\frac{2}{x^{3}}\,dx.

The integral becomes

-\frac{1}{2}\int t^{1/2}\log t\,dt.

Using integration by parts with

u = \log t

and

dv = t^{1/2}dt

\int t^{1/2}\log t\,dt = \frac{2}{3}t^{3/2}\log t - \frac{4}{9}t^{3/2}.

Therefore,

-\frac{1}{2}\int t^{1/2}\log t\,dt = -\frac{1}{3}t^{3/2}\log t + \frac{2}{9}t^{3/2} + c.

Substitute

t = 1+\frac{1}{x^{2}}

\left(1+\frac{1}{x^{2}}\right)^{3/2}\left[-\frac{1}{3}\log\left(1+\frac{1}{x^{2}}\right) + \frac{2}{9}\right] + c.

Answer:

Option A.

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