e^x ( 1 - x 1 - x ) , ,dx is equal to - Vidyadip

MCQ

$\int {{e^x}\left( {\frac{{1 - \sin x}}{{1 - \cos x}}} \right)\,\,dx} $ is equal to

A
$ - {e^x}\tan \,\left( {x/2} \right)$
✓
$ - {e^x}\cot \,\left( {x/2} \right)$
C
$ - \frac{1}{2}{e^x}\tan \,\left( {\frac{x}{2}} \right)$
D
$\frac{1}{2}{e^x}\cot \,\left( {\frac{x}{2}} \right)$

✓

Answer

Correct option: B.

$ - {e^x}\cot \,\left( {x/2} \right)$

b
(b) $I = \int {{e^x}\left( {\frac{{1 - \sin x}}{{1 - \cos x}}} \right)dx} $$ = \int {{e^x}\left( {\frac{{1 - \sin x}}{{2{{\sin }^2}(x2)}}} \right)\,dx} $
==> $I = \int {{e^x}\left( {\frac{1}{2}{\rm{cose}}{{\rm{c}}^2}\frac{x}{2} - \cot \frac{x}{2}} \right)} \,dx$
$\left( \because \,\,\int{{{e}^{x}}\left( f(x)+f'(x) \right)}={{e}^{x}}f(x)+c \right)$
$\therefore \,\,I = {e^x}\left( { - \cot \frac{x}{2}} \right) + c = - {e^x}\cot \frac{x}{2} + c$.

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 "M.C.Q (1 Marks)" from STD 12 - 7.1 inderfinite integral→STD 12 - 7.1 inderfinite integral — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

Choose the correct answer from the given four option.
$\text{y}= \text{a}\text{e}^{\text{mx}}+\text{b}\text{e}^{-\text {mx}}$ satisfies which of the following differential equation?

Let $y=y(x)$ be the solution of the differential equation, $x y^{\prime}-y=x^{2}(x \cos x+\sin x), x>0$ If $y (\pi)=\pi,$ then $y ^{\prime \prime}\left(\frac{\pi}{2}\right)+ y \left(\frac{\pi}{2}\right)$ is equal to

Let $a, b, c$ be distinct non-negative numbers. If the vectors $ai + aj + ck,\,\,i + k$ and $ci + cj + bk$ lie in a plane, then $c$ is

If $f(x) = \left\{ \begin{array}{l}\frac{x}{{{e^{1/x}} + 1}},\,\,{\rm{when\,\,}}\,\,x \ne 0\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,0,\,\,{\rm{when \,\,}}x = 0\end{array} \right.$, then

If vector $\vec{a}=2 \hat{i}+5 \hat{j}$ and $b=2 \hat{i}-\hat{j}$ then unit vector in the direction of the vector $\vec{a}+\vec{b}$ is :

The value of the determinant $\begin{vmatrix} 5 &\text{amp; } 1 \\ 3 &\text{amp; } 2 \end{vmatrix}$

Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if l is perpendicular to m for all l, m ∈ L. Then, R is:

The area bounded by the y-axis, $\text{y}=\cos\text{x}$ and $\text{y}=\sin\text{x}$ when $0\leq\text{x}\leq\frac{\pi}{2}$ is:

If $\vec{a}=4 \hat{i}+6 \hat{j}$ and $\vec{b}=3 \hat{j}+4 \hat{k}$, then the vector form of the component of $\vec{a}$ along $\vec{b}$ is

The distance of the plane $2x - 3y + 6z + 7 = 0$ from the point $(2, -3, -1)$ is: