Evalute the following integrals: a b+ce^x dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{\text{a}}{\text{b}+\text{ce}^\text{x}}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{\text{a}}{\text{b}+\text{ce}^\text{x}}\text{dx}$
Dividing numerator and denomimator by $e^x$
$\Rightarrow\text{I}=\int\frac{\text{ae}^{-\text{x}}}{\text{be}^{-\text{x}}+\text{c}}\text{dx}$
Putting $e^{-x} = t$
$\Rightarrow-\text{e}^{-\text{x}}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\text{e}^{-\text{x}}\text{dx}=-\text{dt}$
$\therefore\text{I}=\int\frac{-\text{a}}{\text{bt}+\text{c}}\text{dt}$
$=\frac{-\text{a}}{\text{b}}\text{ ln}|\text{bt}+\text{c}|+\text{C}$
$\Big[\because\int\frac{1}{\text{ax}+\text{b}}\text{dx}=\frac{1}{\text{a}}\text{ ln}|\text{ax}+\text{b}|+\text{C}\Big]$
$=\frac{-\text{a}}{\text{b}}\text{ ln}|\text{be}^{-\text{x}}+\text{c}|+\text{C}\ \big[\because\text{t}=\text{e}^{-\text{x}}+\text{C}\big]$

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

Similar questions

Construct the composition table for $+_5$ on set $S = \{0, 1, 2, 3, 4\}.$

Find the particular solution of the differential equation $\left( 1 + x ^ { 2 } \right) \frac { d y } { d x } + 2 x y = \frac { 1 } { 1 + x ^ { 2 } }$, given that $y = 0$ when $x = 1$.

If $a_1, a_2, a_3, ...,$ ar are in G.P., then prove that the determinant $\begin{bmatrix}\text{a}_{\text{r}+1}&\text{a}_{\text{r}+5}&\text{a}_{\text{r}+9}\\\text{a}_{\text{r}+7}&\text{a}_{\text{r}+11}&\text{a}_{\text{r}+15}\\\text{a}_{\text{r}+11}&\text{a}_{\text{r}+17}&\text{a} _{\text{r}+21}\end{bmatrix}$ is independent of r.

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\text{a}+\text{x}}{1-\text{ax}}\Big)$

If $\text{x}\begin{bmatrix}2\\3 \end{bmatrix}+\text{y}\begin{bmatrix}-1\\1 \end{bmatrix}=\begin{bmatrix}10\\5 \end{bmatrix},$ find the value of x.

$\text{If x}\sqrt{1+\text{y}}+\text{y}\sqrt{1+\text{x}}=0,$ for, < x < 1, prove that

Find the shortest distance between lines $\vec{r}=6 \hat{i}+2 \hat{j}+2 \hat{k}+\lambda(\hat{i}-2 \hat{j}+2 \hat{k})$ and $\vec{r}=-4 \hat{i}-\hat{k}+\mu(3 \hat{i}-2 \hat{j}-2 \hat{k})$

Differentiate following w.r.t. x:
$\sin^\text{n}\big(\text{ax}^2+\text{bx}+\text{c}\big)$

ABCDE is a pentagon, prove that,$\overrightarrow{\text{AB}}+\overrightarrow{\text{AE}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{DC}}+\overrightarrow{\text{ED}}+\overrightarrow{\text{AC}}=3\ \overrightarrow{\text{AC}}$

Show that the function $f: R \rightarrow R$ given by $f(x) = x^3$ is injective.