If _0^1 1 (5+2 x -2 x ^2 ) (1+ e ^ (2-4 x) ) dx =1 _ e ( +1 ) , >… - Vidyadip

MCQ

If $\int \limits_0^1 \frac{1}{\left(5+2 x -2 x ^2\right)\left(1+ e ^{(2-4 x)}\right)} dx =\frac{1}{\alpha} \log _{ e }\left(\frac{\alpha+1}{\beta}\right)$ $\alpha, \beta > 0$, then $\alpha^4-\beta^4$ is equal to:

✓
$21$
B
$0$
C
$19$
D
$-21$

✓

Answer

Correct option: A.

$21$

a
$I=\int \limits_0^1 \frac{d x}{\left(5+2 x-2 x^2\right)\left(1+ e ^{2-4 \pi}\right)}$

$x \rightarrow 1-x$

$I=\int \limits_0^1 \frac{e^{2-4 x} d x}{\left(5+2 x-2 x^2\right)\left(1+ e ^{2-4 x}\right)}$

Add $(i)$ and $(ii)$

$2 I=\int \limits_0^1 \frac{d x}{5+2 x-2 x^2}=\int \limits_0^1 \frac{d x}{2\left(\frac{11}{4}-\left(x-\frac{1}{2}\right)^2\right)}$

$I=\frac{1}{\sqrt{11}} \ln \left(\frac{\sqrt{11}+1}{\sqrt{10}}\right) \quad \begin{array}{l}\alpha=\sqrt{11} \\\beta=\sqrt{10}\end{array}$

$\alpha^4-\beta^4=121-100=21$

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 - 7.2 definite integral→STD 12 - 7.2 definite integral — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

If $y = x + e^x$ , then at $x = 1$ , $\frac{{{d^2}x}}{{d{y^2}}}$ is equal to

The resultant of two concurrent forces $\vec{\text{nOP}}$ and $\vec{\text{mOQ}}$ is$(\text{m+n})\vec{\text{OR.}}$ Then R divides PQ in the ratio:

If ${\sin ^{ - 1}}\frac{3}{5} + {\cos ^{ - 1}}\left( {\frac{{12}}{{13}}} \right) = {\sin ^{ - 1}}C,$ then $ C =$

Let $f(x) =\left| {\begin{array}{*{20}{c}}
{\cos \,x}&{\sin \,x}&{\cos \,x}\\
{\cos \,2x}&{\sin \,2x}&{2\,\cos \,2x}\\
{\cos \,3x}&{\sin \,3x}&{3\,\cos \,3x}
\end{array}} \right| $then $f ' \left( {\frac{\pi }{2}} \right)$=

Find the value of :$ \sec^2 (\tan^{-1} 2) +\csc^2 (\cot^{-1} 3)$

If a line makes angles $\frac{\pi}{2}, \frac{3 \pi}{4}$ and $\frac{\pi}{4}$ with $X, Y$, and $Z$-axes respectively, then its direction cosines are

Choose the correct option from given four options:
$\int\tan^{-1}\sqrt{\text{x}}\text{ dx}$ is equal to:

The integral $\int {x\,{{\cos }^{ - 1}}\,\left( {\frac{{1 - {x^2}}}{{1 + {x^2}}}} \right)dx} \,\left( {x > 0} \right)$ is equal to

Let $ p, q, r $ be three mutually perpendicular vectors of the same magnitude. If a vector $x$ satisfies equation $p \times \{ (x - q) \times p\} + q \times \{ (x - r) \times q\} + r \times \{ (x - p) \times r\} = 0,$ then $ x$ is given by

The function $\text{f}:\Big[\frac{-1}{2},\frac{1}{2},\frac{1}{2}\Big]\rightarrow\ \Big[\frac{-\pi}{2},\frac{\pi}{2}\Big],$ defined by $\text{f(x)}=\sin^{-1}(3\text{x}-4\text{x}^3),$ is: