Integrate the rational function : 1 (e^x-1 ) - Vidyadip

Question

Integrate the rational function : $\frac{1}{\left(e^x-1\right)}$

✓

Answer

$\begin{array}{l} I =\int \frac{d x}{\left(e^x-1\right)} \\ =\int \frac{e^x}{e^x\left(e^x-1\right)} d x \\ \rightarrow \text { Take, } e^x=t \\ \therefore e^x \cdot d x=d t \\ \text { I }=\int \frac{d t}{t(t-1)} \\ \text { I }=\int \frac{t-(t-1)}{t(t-1)} d t \\ =\int \frac{d t}{t-1}-\int \frac{d t}{t} \\ =\log |t-1|-\log |t|+c\end{array}$
$\begin{array}{l} I =\log \left|e^x-1\right|-\log \left|e^x\right|+c \\ I =\log \left|\frac{e^x-1}{e^x}\right|+c\end{array}$

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 "2 Marks" from JUNE 2024→JUNE 2024 — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the angle betwwen two vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ if

$|\vec{\text{a}}|=3,\big|\vec{\text{b}}\big|=3$ and $\vec{\text{a}}.\vec{\text{b}}=1$

Evaluate the following integrals:
$\int\text{xe}^\text{x}\text{dx}$

Write the coordinates of the projection of point P(2, -3, 5) on Y-axis.

Find the volume of the parallelopiped with its edges represented by the vectors
$\hat{\text{i}}+\hat{\text{j}},\hat{\text{i}}+2\hat{\text{j}}$ and $\hat{\text{i}}+\hat{\text{j}}+\pi{\text{k}}.$

Evaluate the following integrals:
$\int\text{x}^{\frac{5}{4}}\text{dx}$

Find the value of ${\tan ^{ - 1}}\left[ {2\cos \left( {2{{\sin }^{ - 1}}\left( {\frac{1}{2}} \right)} \right)} \right]$.

Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

If $\text{f}'\text{(x)}=\sqrt{2\text{x}^2-1}$ and $\text{y}=\text{f}(\text{x}^2),$ then find $\frac{\text{dy}}{\text{dx}}\text{at x}=1.$

Evaluate: $\begin{vmatrix}\cos15^\circ&\sin15^\circ\\\sin75^\circ&\cos75^\circ\end{vmatrix}$

If $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\ \vec{\text{b}}=4\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}}$and $\vec{\text{c}}=\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}$, find a vecctor of magnitude 6 units which is parallel to the vector $2\vec{\text{a}}-\vec{\text{b}}+3\vec{\text{c}}$.