Mathematics

Integrate: $\int \dfrac{x}{x^4-x^2+1}\ dx$

SOLUTION
Solution:-
$\int{\cfrac{x}{{x}^{4} - {x}^{2} + 1} dx}$
Let ${x}^{2} = t \Rightarrow x \; dx = \cfrac{dt}{2}$
$\therefore \int{\cfrac{x}{{x}^{4} - {x}^{2} + 1} dx} = \cfrac{1}{2} \int{\cfrac{dt}{{t}^{2} - t + 1}}$
$\Rightarrow = \cfrac{1}{2} \int{\cfrac{dt}{{\left( t - \cfrac{1}{2} \right)}^{2} - \cfrac{1}{4} + 1}}$
$\Rightarrow = \cfrac{1}{2} \int{\cfrac{dt}{{\left( t - \cfrac{1}{2} \right)}^{2} + {\left( \cfrac{\sqrt{3}}{2} \right)}^{2}}}$
$\Rightarrow = \cfrac{1}{2} \left[ \dfrac{1}{\left( \cfrac{\sqrt{3}}{2} \right)} \tan^{-1}{\dfrac{\left( t - \dfrac{1}{2} \right)}{\left( \cfrac{\sqrt{3}}{2} \right)}} \right] + C$
$\Rightarrow = \cfrac{\sqrt{3}}{4} \left( \tan^{-1}{\left( \cfrac{2t - 1}{\sqrt{3}} \right)} \right) + C$
As $t = {x}^{2}$
$\therefore \int{\cfrac{x}{{x}^{4} - {x}^{2} + 1} dx} = \cfrac{\sqrt{3}}{4} \left( \tan^{-1}{\left( \cfrac{2{x}^{2} - 1}{\sqrt{3}} \right)} \right) + C$

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

Realted Questions

Q1 Multiple Correct Hard
$\int_{0}^{2\pi}sin^{4}$ x dx is equal to
• A. $8\int_{0}^{\frac{\pi}{4}}sin^{4}$ x dx
• B. $3\int_{0}^{\frac{2\pi}{3}}sin^{4}$ x dx
• C. $2\int_{0}^{\pi}sin^{4}$ x dc
• D. $4\int_{0}^{\frac{\pi}{2}}cos^{4}$ x dx

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Solve $I = \displaystyle\int\dfrac{1}{\cos^2x(1-\tan x)^2}dx$.
• A. $I=\cfrac{-1}{1 -\cot x}+C$
• B. $I=\cfrac{1}{1 -\tan x}+C$
• C. None of these
• D. $I=\cfrac{-1}{1 -\tan x}+C$

1 Verified Answer | Published on 17th 09, 2020

Q3 One Word Medium
Number of real solution of the given equation for $x$, $\int x^{2}\ e^{x}dx=0$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\underset{-a}{\overset{a}{\int}} \sqrt{\dfrac{a - x}{a + x}} dx$ is equal to
• A. $\pi$
• B. a
• C. $\dfrac{a \pi}{2}$
• D. $a \pi$

Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.