_0^1 e^ - x 1 + e^ - x ,dx = - Vidyadip

MCQ

$\int_0^1 {\frac{{{e^{ - x}}}}{{1 + {e^{ - x}}}}} \,dx = $

A
$\log \left( {\frac{{1 + e}}{e}} \right) - \frac{1}{e} + 1$
✓
$\log \left( {\frac{{1 + e}}{{2e}}} \right) - \frac{1}{e} + 1$
C
$\log \left( {\frac{{1 + e}}{{2e}}} \right) + \frac{1}{e} - 1$
D
None of these

✓

Answer

Correct option: B.

$\log \left( {\frac{{1 + e}}{{2e}}} \right) - \frac{1}{e} + 1$

b
(b) Put $1 + {e^{ - x}} = t $

$\Rightarrow - {e^{ - x}}dx = dt$, then we have

$I = \int_2^{1 + \frac{1}{e}} {\frac{{(t - 1)( - dt)}}{t}} $

$= \int_2^{1 + \frac{1}{e}} {\left( {\frac{1}{t} - 1} \right)} \,dt$

$ = \left[ {{{\log }_e}t - t} \right]_2^{1 + \frac{1}{e}} $

$= {\log _e}\left( {1 + \frac{1}{e}} \right) - \left( {1 + \frac{1}{e}} \right) - {\log _e}2 + 2$

$ = {\log _e}\left( {\frac{{e + 1}}{{2e}}} \right) - \frac{1}{e} + 1$.

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 7.2 definite integral→7.2 definite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

$\begin{bmatrix}2\text{x}&5\\8&\text{x}\end{bmatrix}=\begin{bmatrix}6\text{x}&-2\\7&3\end{bmatrix},$ then the value of $\text{x}$ is:

$3$
$\pm3$
$\pm6$
$6$

Choose the correct answer from the given four options.
Let us define a relation R in R as aRb if a ≥ b. Then R is:

An equivalence relation.
Reflexive, transitive but not symmetric.
Symmetric, transitive but not reflexive.
Neither transitive nor reflexive but symmetric.

If $f(x) = \left\{ \begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,x\sin x,\,{\rm{when \,\,}}0 < x \le \frac{\pi }{2}\\\frac{\pi }{2}\sin (\pi + x),{\rm{when\,\,}}\frac{\pi }{{\rm{2}}} < x < \pi \end{array} \right.$, then

Let $A= \{1, 2, 3, 4\}$ and $R : A \to A$ be the relation defined by $R = \{ (1, 1), (2, 3), (3, 4), ( 4, 2) \}$. The correct statement is

If ${\tan ^{ - 1}}x + {\tan ^{ - 1}}y = \frac{\pi }{4}$ then

If the position vectors of P, Q are respectively 5a + 4b and 3a - 2b then $\vec{\text{QP}}=$

2a + 6b
2a − 6b
2a + 5b
2a − 5b

The sum of squares of all possible values of $k$, for which area of the region bounded by the parabolas $2 \mathrm{y}^2=\mathrm{kx}$ and $\mathrm{ky}^2=2(\mathrm{y}-\mathrm{x})$ is maximum, is equal to :

Region represented by $x \geq 0, y \geq 0$ is

Let $M$ and $N$ be two $3 \times 3$ non-singular skew-symmetric matrices such that $M N=N M$. If $P^T$ denotes the transpose of $P$, then $M^2 N^2\left(M^T N\right)^{-1}\left(M N^{-1}\right)^T$ is equal to

$(A)$ $M^2$ $(B)$ $-N^2$ $(C)$ $-M^2$ $(D)$ $M N$

If a * b denote the bigger among a and b and if ab = (a * b) + 3, then 4.7 =

14
31
10
8