Mathematics

# The value of $\int {{x \over {\sqrt {{x^4} + {x^2} + 1} }}dx}$ equals

${1 \over 2}\log \left( {\left( {{x^2} + {1 \over 2}} \right) + \sqrt {{x^4} + {x^2} + 1} } \right) + c$

##### SOLUTION
we  have to evaluate $I =\int\dfrac{x}{\sqrt{x^4+ x^2 +1}}dx$

Let us assume $x^2 = t\Rightarrow 2 x dx=dt\Rightarrow xdx = \dfrac{dt}{2}$

then,
$\Rightarrow I=\dfrac12\int\dfrac{1}{\sqrt{t^2 +t+1}}dt$

$\Rightarrow I=\dfrac12\int\dfrac{1}{\sqrt{t^2 +t+\dfrac14-\dfrac14+1}}dt$

$\Rightarrow I=\dfrac12\int\dfrac{1}{\sqrt{(t+\dfrac12)^2+ (\dfrac{\sqrt3}{2})^2}}dt$

$\Rightarrow \dfrac12log\begin{pmatrix}(t+\dfrac12)+\sqrt{(t+\dfrac12)^2+(\dfrac{\sqrt3}{2})^2}\end{pmatrix}$

from
$\begin{bmatrix}\because \int\dfrac{1}{\sqrt{a^2+x^2}}dx = log\begin{bmatrix} x+ \sqrt{x^2+a^2}\end{bmatrix}+ C\end{bmatrix}$

again put $t= x^2$

$\Rightarrow \dfrac12log\begin{pmatrix}(x^2+\dfrac12)+\sqrt{(x^2+\dfrac12)^2+(\dfrac{\sqrt3}{2})^2}\end{pmatrix} + C$

$= \dfrac12log\begin{pmatrix}(x^2+\dfrac12)+\sqrt{(x^4+x^2+1}\end{pmatrix} + C$

Hence Correct answer is $B$.

Its FREE, you're just one step away

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

#### Realted Questions

Q1 Single Correct Hard
Which of the following functions does not appear in the primitive of $\dfrac {dx}{x+\sqrt {{x}^{2}-x+1}}$ if $t$ is a function of $x$?
• A. ${\log}_{e}|t-2|$
• B. ${\log}_{e}|t-1|$
• C. ${\log}_{e}|t+1|$
• D. ${\log}_{e}|t|$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\int_{-1/2}^{1/2} ( \cos x) \left[ log \left( \dfrac{1-x}{1+x} \right) \right] dx$ is equal to :
• A. $1$
• B. $e^{1/2}$
• C. $2e^{1/2}$
• D. $0$

1 Verified Answer | Published on 17th 09, 2020

Q3 Assertion & Reason Hard
##### ASSERTION

Statement 1:$\displaystyle F(x)=\int_{1}^{x}\frac{\log t}{1+t+t^{2}}dt$ then $F(x)=-F(1/x)$

##### REASON

Statement 2: If $\displaystyle F(x)=\int_{1}^{x}\frac{\log t}{t+1}dt$ then $F(x)+F(1/x)=(1/2)(\log x)^{2}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

1 Verified Answer | Published on 17th 09, 2020

Q4 Multiple Correct Hard
If $\displaystyle I_{n}=\int_{-\pi }^{\pi }\frac{\sin nx}{(1+\pi ^{x})\cdot \sin x}dx,n=0,1,2,...,$then
• A. $\displaystyle I_{n}=I_{n+1}$
• B. $\displaystyle I_{n}=I_{n+2}$
• C. $\displaystyle \sum_{m=1}^{10}I_{2m+1}=10\pi$
• D. $\displaystyle \sum_{m=1}^{10}I_{2m}=0$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$