_ ^ dx x[ ( x) ^2 + 4 x - 1] = - Vidyadip

MCQ

$\int_{}^{} {\frac{{dx}}{{x[{{(\log x)}^2} + 4\log x - 1]}}} = $

✓
$\frac{1}{{2\sqrt 5 }}\log \left[ {\frac{{\log x + 2 - \sqrt 5 }}{{\log x + 2 + \sqrt 5 }}} \right] + c$
B
$\frac{1}{{\sqrt 5 }}\log \left[ {\frac{{\log x + 2 - \sqrt 5 }}{{\log x + 2 + \sqrt 5 }}} \right] + c$
C
$\frac{1}{{2\sqrt 5 }}\log \left[ {\frac{{\log x + 2 + \sqrt 5 }}{{\log x + 2 - \sqrt 5 }}} \right] + c$
D
$\frac{1}{{\sqrt 5 }}\log \left[ {\frac{{\log x + 2 + \sqrt 5 }}{{\log x + 2 - \sqrt 5 }}} \right] + c$

✓

Answer

Correct option: A.

$\frac{1}{{2\sqrt 5 }}\log \left[ {\frac{{\log x + 2 - \sqrt 5 }}{{\log x + 2 + \sqrt 5 }}} \right] + c$

a
(a) Put $\log x = t \Rightarrow \frac{1}{x}\,dx = dt,$ then
$\int_{}^{} {\frac{{dx}}{{x[{{(\log x)}^2} + 4\log x - 1]}}} = \int_{}^{} {\frac{{dt}}{{{t^2} + 4t - 1}}} $
$ = \int_{}^{} {\frac{{dt}}{{{{(t + 2)}^2} - {{(\sqrt 5 )}^2}}} = \frac{1}{{2\sqrt 5 }}\log \left[ {\frac{{t + 2 - \sqrt 5 }}{{t + 2 + \sqrt 5 }}} \right]} $
$ = \frac{1}{{2\sqrt 5 }}\log \left[ {\frac{{\log x + 2 - \sqrt 5 }}{{\log x + 2 + \sqrt 5 }}} \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 "M.C.Q (1 Marks)" from STD 12 - 7.1 inderfinite integral→STD 12 - 7.1 inderfinite integral — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

$\text{y}=\sin kt$ satisfies the differential equation $y\ "+\ 9y = 0$. Then $k$ :

Choose the correct answer from the given four options.
Distance of the point $(\alpha,\beta,\gamma)$ from y-axis is:

The solution of the differential equation $\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}+\text{x}\ \tan\frac{\text{y}}{\text{x}}$ is:

Let , $f(x) = \min \left\{ {{{\sin }^{ - 1}}x,{{\cos }^{ - 1}}x} \right\}$ then area bounded by $f(x)$ and $x-$ axis is

The order and degree of the differential equation $\Big(1+3\frac{\text{dy}}{\text{dx}}\Big)^\frac{2}{3}=4\frac{\text{d}^3\text{y}}{\text{dx}^3}$ are:

Let $f :[2,4] \rightarrow R$ be a differentiable function such that $\left(x \log _e x\right) f^{\prime}(x)+\left(\log _e x\right) f(x)+f(x) \geq 1$, $x \in[2,4]$ with $f(2)=\frac{1}{2}$ and $f(4)=\frac{1}{4}$.

Consider the following two statements:

$(A): f(x) \leq 1$, for all $x \in[2,4]$

$(B)$ : $f(x) \geq \frac{1}{8}$, for all $x \in[2,4]$

Then,

The area bounded by the curve $x = 3y^2 – 9$ and the line $x = 0, y = 0$ and $y = 1$ is:

Find the values of $x, y, z$ and $w$ respectively such that $\left[\begin{array}{cc}x-y & 2 z+w \\ 2 x-y & 2 x+w\end{array}\right]=\left[\begin{array}{cc}5 & 3 \\ 12 & 15\end{array}\right]$.

A person throws two fair dice. He wins $Rs.\, 15$ for throwing a doublet (same numbers on the two dice), wins $Rs.\,12$ when the throw results in the sum of $9$, and loses $Rs.\, 6$ for any other outcome on the throw. Then the expected gain/loss (in $Rs.$) of the person is

The general solution $y(x)$ of the differential equation $\frac{{d\left( {\int\limits_x^y {dt} } \right)}}{{dy}} = x$ , is