Integrate the function x^ 2 +1 [ (x^ 2 +1 )-2 x ] x^ 4

Question

Integrate the function $\frac{\sqrt{x^{2}+1}\left[\log \left(x^{2}+1\right)-2 \log x\right]}{x^{4}}$

✓

Answer

Given integrand is: $\frac{\sqrt{x^{2}+1}\left[\log \left(x^{2}+1\right)-2 \log x\right]}{x^{4}}$
Here we can rewrite integrand as; $ \frac{\sqrt{x^{2}+1}\left[\log \left(x^{2}+1\right)-2 \log x\right]}{x^{4}}$
= $\frac{\sqrt{x^{2}+1}}{x^{4}}\left[\log \left(x^{2}+1\right)-\log x^{2}\right]$
= $\frac{1}{x^{3}} \sqrt{\frac{x^{2}+1}{x^{2}}}\left[\log \left(\frac{x^{2}+1}{x^{2}}\right)\right]$
= $\frac{1}{x^{3}} \sqrt{1+\frac{1}{x^{2}}}\left[\log \left(1+\frac{1}{x^{2}}\right)\right]$
Let, $1+\frac{1}{x^{2}}=t \Rightarrow-\frac{2}{x^{3}} d x=d t$
Now, Let $I = \int \frac{\sqrt{x^{2}+1}\left[\log \left(x^{2}+1\right)-2 \log x\right]}{x^{4}} d x$ = $\int \frac{1}{x^{3}} \sqrt{1+\frac{1}{x^{2}}}\left[\log \left(1+\frac{1}{x^{2}}\right)\right] d x$
= $\int-\frac{1}{2} \cdot \sqrt{t}[\log (t)] d t$
= $\int-\frac{1}{2} \cdot \sqrt{t}[\log (t)] d t$ = $-\frac{1}{2}\left[\log \mathrm{t} . \int \sqrt{\mathrm{t}} \mathrm{d} \mathrm{t}-\int \frac{\mathrm{d}}{\mathrm{dt}} \log \mathrm{t} .\left\{\int \sqrt{\mathrm{t}} \mathrm{d} \mathrm{t}\right\} \mathrm{dt}\right]$
= $-\frac{1}{2}\left[\log \mathrm{t} \cdot \frac{\mathrm{t}^{\frac{3}{2}}}{\frac{3}{2}}-\int \frac{1}{\mathrm{t}} \cdot\left\{\frac{\mathrm{t}^{\frac{3}{2}}}{\frac{3}{2}}\right\} \mathrm{dt}\right]$
= $-\frac{1}{2}\left[\frac{2}{3} t^{\frac{3}{2}} \log t-\int\left\{\frac{t^{\frac{1}{2}}-1}{\frac{3}{2}}\right\} d t\right]$
= $-\frac{1}{2}\left[\frac{2}{3} t^{\frac{3}{2}} \log t-\frac{2}{3} \int t^{\frac{1}{2}} d t\right]$
= $-\frac{1}{2}\left[\frac{2}{3} t^{\frac{3}{2}} \log t-\frac{2}{3} \cdot \frac{t^{\frac{3}{2}}}{\frac{3}{2}}\right]$
= $\left[-\frac{1}{2} \cdot \frac{2}{3} t^{\frac{3}{2}} \log t+\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{2}{3} \cdot t^{\frac{3}{2}}\right]$
= $-\frac{1}{3} t^{\frac{3}{2}}\left[\log t-\frac{2}{3}\right]$
$\Rightarrow 1=-\frac{1}{3}\left(1+\frac{1}{x^{2}}\right)^{\frac{3}{2}}\left[\log \left(1+\frac{1}{x^{2}}\right)-\frac{2}{3}\right]+C$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "1 Marks Question" from Integrals→Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Integrals — MATHS STD 12 Science — Question

Answer

Need a full question paper?

Explore more

Similar questions