Evaluate the following integrals: e^ 2x 1+e^x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\text{e}^{2\text{x}}}{1+\text{e}^\text{x}}\text{ dx}$

✓

Answer

$\int\frac{\text{e}^{2\text{x}}}{1+\text{e}^\text{x}}\text{ dx}$ $\Rightarrow\int\frac{\text{e}^\text{x}.\text{e}^\text{x}}{1+\text{e}^\text{x}}\text{ dx}$ then, Let $1+\text{e}^\text{x}=\text{t}$ $\Rightarrow\text{e}^\text{x}=\text{t}-1$ $\Rightarrow\text{e}^\text{x}\text{dx}=\text{dt}$ Now, $\int\frac{\text{e}^\text{x}.\text{e}^\text{x}}{1+\text{e}^\text{x}}\text{ dx}$$=\int\frac{(\text{t}-1)\text{dt}}{\text{t}}$
$=\Big(1-\frac{1}{\text{t}}\Big)\text{dt}$
$=\text{t}-\log|\text{t}|+\text{C}$
$=\big(1+\text{e}^\text{x}\big)-\log\big(1+\text{e}^\text{x}\big)+\text{C}$
Let $\text{C}+1=\text{C}^\text{n}$ $=\text{e}^\text{x}-\log\big(1+\text{e}^\text{x}\big)+\text{C}^\text{n}$

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 "Solve the Following Question.(3 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\frac{1}{\text{x}^2+6\text{x}+13}\text{dx}$

A coin is tossed three times, if head occurs on first two tosses, find the probability of getting head on third toss.

In $\triangle \mathrm{ABC}$, prove that $\sin \left(\frac{\mathbf{B}-\mathbf{C}}{2}\right)=\left(\frac{\boldsymbol{b}-\boldsymbol{c}}{a}\right) \cos \frac{A}{2}$.

Find $m$, if the lines $\frac{1-x}{3}=\frac{7 y-14}{2 m }=\frac{z-3}{2}$ and $\frac{7-7 x}{3 m }=\frac{y-5}{1}=\frac{6-z}{5}$ are at right angles

Determine which of the following binary operations are associative and which are commutative:
* on Q defined by $\text{a}\ ^*\ \text{b}=\frac{\text{a}+\text{b}}{2}$ for all $\text{a, b}\in\text{Q}$

Let $\bar{a}$ and $\bar{b}$ be non-collinear vectors. If vector $\bar{r}$ is co-planar with $\bar{a}$ and $\bar{b}$ then prove that there exists unique scalars $t_1$ and $t_2$ such that $\bar{r}=t_1 \bar{a}+t_2 \bar{b}$. Hence find $t_1$ and $t_2$ for $\bar{r}=$ $\hat{\imath}+\hat{\jmath}, \bar{a}=2 \hat{\imath}-\hat{\jmath}, \bar{b}=\hat{\imath}-2 \hat{\jmath}$.

Find the vector and cartesian equations of a plane passing through the point (1, -1, 1) and normal to the line joining the points (1, 2, 5) and (-1, 3, 1)

Solve the following differential equations.

$\frac{\cos ^2 y}{x} d y+\frac{\cos ^2 x}{y} d x=0$

Evaluate the following integrals:
$\int\frac{1-\cos2\text{x}}{1+\cos2\text{x}}\text{dx}$

Find the mean and standard deviation of the following probability distributions:

$x_i$	-3	-1	0	1	3
$p_i$	0.05	0.45	0.20	0.25	0.05