Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS1 Mark
MCQ
Choose the correct answer in exercise$:\ \int\frac{\text{dx}}{\text{x}(\text{x}^2+1)}$ equals
  • ✓
    $\log|x|-\frac{1}{2}\log(\text{x}^2+1)+\text{C}$
  • B
    $\log|x|+\frac{1}{2}\log(\text{x}^2+1)+\text{C}$
  • C
    $-\log|\text{x}|+\frac{1}{2}\log(\text{x}^2+1)+\text{C}$
  • D
    $\log|\text{x}|+\frac{1}{2}\log(\text{x}^2+1)+\text{C}$
✓

Answer

Correct option: A.
$\log|x|-\frac{1}{2}\log(\text{x}^2+1)+\text{C}$
Let $\text{I}=\int\frac{1}{\text{x}(\text{x}^2+1)}\text{dx}=\int\frac{2\text{x}}{2\text{x}^2(\text{x}^2+1)}\text{dx}=\int\frac{2\text{x}}{2\text{x}^2(\text{x}^2+1)}\text{dx}\dots(\text{i})$
Putting $x^2 = t$
$\Rightarrow 2x dx = dt$
Putting this value in $\text{eq. (i),}$
$\text{I}=\int\frac{\text{dt}}{2\text{t}(\text{t}+1)}=\frac{1}{2}\int\frac{(\text{t}+1)-\text{t}}{\text{t}(\text{t}+1)}\text{dt}$
$\text{I}=\frac{1}{2}\int\Bigg(\frac{\text{t}+1}{\text{t}(\text{t}+1)}-\frac{\text{t}}{\text{t}(\text{t}+1)}\Bigg)\text{dt}=\frac{1}{2}\int\Bigg(\frac{1}{\text{t}}-\frac{1}{\text{t}+1}\Bigg)\text{dt}=\frac{1}{2}\Bigg[\int\frac{1}{\text{t}}\text{dt}-\int\frac{1}{\text{t}+1}\text{dt}\Bigg]$
$=\frac{1}{2}[\log|\text{t}|-\log|\text{t}+1|]+\text{c}$
$=\frac{1}{2}[\log|\text{x}^2|-\log(\text{x}^2+1)]+\text{c}$
$=\log|\text{x}|-\frac{1}{2}\log|\text{x}^2+1|+\text{c}$

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 "M.C.Q (1 Marks)" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

An orthogonal matrix is
→
Let $f$ and $g$ be two functions defined by

$f(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\|x-1|, & x \geq 0\end{array} \text { and } g(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\1, & x \geq 0\end{array}\right. \text {. }\right.$

Then (gof) (x) is

→
If f : A → B is surjective then:
→
The area bounded by the curve $\text{y}=\frac{3}{2}\sqrt{\text{x}},$ the line $\text{x}=1$ and $x -$ axis is $..........sq.$ units:
→
Let $S=\left\{ w _1, w _2, \ldots.\right\}$ be the sample space associated to a random experiment. Let $P \left( w _{ n }\right)=\frac{ P \left( w _{ n -1}\right)}{2}, n \geq 2$.Let $A=\{2 k +3 \ell ; k , \ell \in N \}$ and $B=\left\{ W _{ n } ; n \in A \right\}$.Then $P ( B )$ is equal to
→
If in a $\triangle\text{ABC}$, $\text{A}=(0,0),\ \text{B}=(3,3\sqrt3),\ \text{C}=(-3\sqrt3,3)$, then the vecctor of magnitude $2\sqrt2$ units directed along AO, where O is the circumcenter of $\triangle\text{ABC}$ is,
→
Let the curve $y = y ( x )$ be the solution of the differential equation, $\frac{ dy }{ dx }=2( x +1) .$ If the numerical value of area bounded by the curve $y = y ( x )$ and $x-$ axis is $\frac{4 \sqrt{8}}{3},$ then the value of $y (1)$ is equal to
→
What are the order and degree, respectively, of the differential equation : $\Big(\frac{\text{d}^3\text{y}}{\text{dx}^3}\Big)^2=\text{y}^4+\Big(\frac{\text{dy}}{\text{dx}}\Big)^5\ ?$
→
Which of the following holds true for a vector quantity:
→
Let $f (x)$ and $g (x)$ be two differentiable function in $R$ and $f(2) = 8, g (2) = 0, f (4) = 10$ and $g (4) = 8$ then
→