Evalute the following integrals: e^ 2x e^ 2x -2 dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{\text{e}^{2\text{x}}}{\text{e}^{2\text{x}}-2}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{\text{e}^{2\text{x}}}{\text{e}^{2\text{x}}-2}\text{dx}$
Putting e^2x = t
$\Rightarrow2\text{e}^{2\text{x}}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\text{e}^{2\text{x}}\text{dx}=\frac{\text{dt}}{2}$
$\therefore\text{I}=\frac{1}{2}\int\frac{1}{\text{t}-2}\text{dt}$
$=\frac{1}{2}\text{ ln}|\text{t}-2|+\text{C}$
$=\frac{1}{2}\text{ ln}\big|\text{e}^{2\text{x}}-2\big|+\text{C }\big[\text{t}=\text{e}^{2\text{x}}\big]$

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" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Show that if $A=\left[\begin{array}{ll} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right], \text { then } A^{n}=\left[\begin{array}{cc} \cos n \theta & \sin n \theta \\ -\sin n \theta & \cos n \theta \end{array}\right]$

Find the volume of the parallelopiped whose coterminous edges are represented by the vectors:
$\vec{\text{a}}=2\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}},\vec{\text{c}}=3\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}$

Evaluate the following integrals:
$\int\frac{1+\cos\text{x}}{(\text{x}+\sin\text{x})^3}\text{dx}$

For any two vectors $\vec{\text{a}}$ and $\vec{\text{b}},$ prove that $\big|\vec{\text{a}}\times\vec{\text{b}}\big|^2=\begin{vmatrix}\vec{\text{a}}.\vec{\text{a}}&\vec{\text{a}}.\vec{\text{b}}\\\vec{\text{b}}.\vec{\text{a}}&\vec{\text{b}} .\vec{\text{b}}\end{vmatrix}.$

Evaluate the following integrals:

$\int\text{e}^{\text{x}}\big[\sec\text{x}+\log(\sec\text{x}+\tan\text{x})\big]\text{dx}$

Express the following matrices as the sum of a symmentric and a skew symmentric matrix:

$\begin{bmatrix}3&5\\1&-1\end{bmatrix}$

Classify the following functions as injection, surjection or bijection:
f : R → R, defined by f(x) = sin²x + cos²x

Find the position vector of a point R which divides the line joining the two points P and Q with position vectors $\overrightarrow{\text{OP}}=2\vec{\text{a}}+\vec{\text{b}}\text{ and }\overrightarrow{\text{OQ}}=\vec{\text{a}}-2\vec{\text{b}}$, respectively in the ratio 1 : 2 internally and externally.

Find the intervals in which the function f given by f(x) = $ x ^ { 3 } + \frac { 1 } { x ^ { 3 } } , x \neq 0$ is x (i) increasing. (ii) decreasing.

Using determinants, find the value of k so that the points (k, 2 - 2k), (-k + 1, 2k) and (-4 - k, 6 - 2k) may be collinear.