_ ^ e^x (1 + e^x )(2 + e^x ) dx = - Vidyadip

MCQ

$\int_{}^{} {\frac{{{e^x}}}{{(1 + {e^x})(2 + {e^x})}}dx = } $

A
$\log [(1 + {e^x})(2 + {e^x})] + c$
✓
$\log \left[ {\frac{{1 + {e^x}}}{{2 + {e^x}}}} \right] + c$
C
$\log [(1 + {e^x})\sqrt {2 + {e^x}} ] + c$
D
None of these

✓

Answer

Correct option: B.

$\log \left[ {\frac{{1 + {e^x}}}{{2 + {e^x}}}} \right] + c$

b
(b)$\int_{}^{} {\frac{{{e^x}}}{{(1 + {e^x})(2 + {e^x})}}\,dx} = \int_{}^{} {\left\{ {\frac{{{e^x}}}{{1 + {e^x}}} - \frac{{{e^x}}}{{2 + {e^x}}}} \right\}dx} $
Now put $1 + {e^x} = t$ and $2 + {e^x} = t,$ then the required integral $ = \log (1 + {e^x}) - \log (2 + {e^x}) = \log \left( {\frac{{1 + {e^x}}}{{2 + {e^x}}}} \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

$ \begin{bmatrix}1 & \text{x} & \text{x}^2 \\1 & \text{y} & \text{y}^2 \\1 & \text{z} & \text{z}^2\end{bmatrix}$

$\tan^{-1}1+\cos^{-1}\Big(\frac{-1}{2}\Big)+\sin^{-1}\Big(\frac{-1}{2}\Big)$

Which of the following statements about an $LP$ problem and its dual is false?

If $A$ and $B$ are square matrices of the same order, then $(A + B)(A - B)$ is equal to:

$\int_{}^{} {\frac{{\sin x\;dx}}{{{a^2} + {b^2}{{\cos }^2}x}}} = $

Choose the correct answer from the given four options.
The probability distribution of a discrete random variable $X$ is given below:

$\text{X}$	$2$	$3$	$4$	$5$
$\text{P}(\text{X})$	$\frac{5}{\text{k}}$	$\frac{7}{\text{k}}$	$\frac{9}{\text{k}}$	$\frac{11}{\text{k}}$

The value of $k$ is:

Choose the correct answer from the given four options.If matrix $A = [a_{ij}]_{2\times 2},$ where $a_{ij} = 1,$ if $\text{i}\neq\text{j}$ and $0$ if $i = j$ then $A^2$ equal to:

Order and degree of the differential equation $\left(1+\left(\frac{d y}{d x}\right)^3\right)^{\frac{7}{3}}=7 \frac{d^2 y}{d x^2}$ are respectively

Let $f(x) = (x - a)^2+ (x - b)^2 + (x - c)^2.$ Then $,f(x)$ has a minimum at $x =$

Let $f$ and $g$ be continuous functions on $[0, a]$ such that $f(x) = f(a -x)$ and $g(x) + g(a -x) = 4$, then $\int\limits_0^a {f\left( x \right)g\left( x \right)dx} $ is equal to