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 "3 Marks Question" from Indefinite Integrals→Indefinite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Solve the differential equation: $\left( x ^3+ x ^2+ x +1\right) \frac{d y}{d x}=2 x ^2+ x$

Evaluate the following integrals:
$\int (3\text{x}\sqrt{\text{x}}+4\sqrt{\text{x}}+5)\text{dx}$

Evaluate the following integrals:
$\int\frac{1}{\text{x}}(\log\text{x})^2\text{dx}$

Show that the vectores $\vec{\text{a}}=3\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}-3\hat{\text{j}}+5\hat{\text{k}},\vec{\text{c}}=2\hat{\text{i}}+\hat{\text{j}}-4\hat{\text{k}}$from a right-angled triangle.

If $\text{A}=\begin{bmatrix}2&3\\5&7\end{bmatrix},\text{ B}=\begin{bmatrix}-1&0&2\\3&4&1\end{bmatrix},\text{C}=\begin{bmatrix}-1&2&3\\2&1&0\end{bmatrix},$ find2B + 3A and 3C - 4B.

Find the value of $\lambda$ so that the following vectors are coplanar:
$\vec{\text{a}}=2\hat{\text{i}}-\hat{\text{j}}++\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}},\vec{\text{c}}=\lambda\hat{\text{i}}+\lambda\hat{\text{j}}+5\hat{\text{k}}$

Find the probability distribution of Y in two throws of two dice, where Y represents the number of times a total of 9 appears.

Find the unit vector in the direction of the resultant of the vectors $\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}},\ 2\hat{\text{i}}+\hat{\text{j}}-2\hat{\text{k}}$ and $\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}$.

For the principal values, evaluate the following:
$\sin^{-1}[\cos\{2\text{cosec}^{-1}(-2)\}]$

Write the value of $\cos^{-1}(\cos350^\circ)-\sin^{-1}(\sin350^\circ)$