Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS1 Mark
MCQ
$\int\text{e}^{\text{x}}\Big(\frac{1-\sin\text{x}}{1-\cos\text{x}}\Big)\text{dx}=$
  • A
    $-\text{e}^{\text{x}}\tan\frac{\text{x}}{2}+\text{C}$
  • ✓
    $-\text{e}^{\text{x}}\cot\frac{\text{x}}{2}+\text{C}$
  • C
    $-\frac{1}{2}\text{e}^{\text{x}}\tan\frac{\text{x}}{2}+\text{C}$
  • D
    $-\frac{1}{2}\text{e}^{\text{x}}\cot\frac{\text{x}}{2}+\text{C}$
✓

Answer

Correct option: B.
$-\text{e}^{\text{x}}\cot\frac{\text{x}}{2}+\text{C}$
Let $\text{I}=\int\text{e}^{\text{x}}\Big(\frac{1-\sin\text{x}}{1-\cos\text{x}}\Big)\text{dx}$

$\int\text{e}^{\text{x}}\Big(\frac{1}{1-\cos\text{x}}-\frac{\sin\text{x}}{1-\cos\text{x}}\Big)\text{dx}$

$\int\text{e}^{\text{x}}\bigg(\frac{1}{2\sin^2\frac{\text{x}}{2}}-\frac{2\sin\frac{\pi}{2}\cos\frac{\pi}{2}}{2\sin^2\frac{\pi}{2}}\bigg)\text{dx}$

$\int\text{e}^\text{x}\Big(\frac{1}{2}\text{cosec}^2\frac{\text{x}}{2}-\cot\frac{\text{x}}{2}\Big)\text{dx}$

As, we know that $\int\text{e}^{\text{x}}\{\text{f(x)}+\text{f}'(\text{x})\}\text{dx}=\text{e}^{\text{x}}\text{f(x)}+\text{C}$

$\therefore\ \text{I}=-\text{e}^{\text{x}}\cot\Big(\frac{\text{x}}{2}\Big)+\text{C}$

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 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

If $f: \mathrm{R} \rightarrow \mathrm{R}$ is a differentiable function such that $f^{\prime}(x)>2 f(x)$ for all $x \in \mathrm{R}$, and $f(0)=1$, then

$[A]$ $f(x)$ is increasing in $(0, \infty)$

$[B]$ $f(x)$ is decreasing in $(0, \infty)$

$[C]$ $f(x)>e^{2 x}$ in $(0, \infty)$

$[D]$ $f^{\prime}(x) < e^{2 x}$ in $(0, \infty)$

→
Let $f(x) ={\left( {{x^x}} \right)^x}$ and $g(x) ={x^{\left( {{x^x}} \right)}}$ then :
→
If $I=\sum_{k=1}^{98} \int_k^{k+1} \frac{k+1}{x(x+1)} d x$, then $[A]$ $I>\log _e 99$   $[B]$ $I<\log _e 99$   $[C]$ $I<\frac{49}{50}$   $[D]$ $I>\frac{49}{50}$
→
If $\text{g}'(\text{x})=\int\text{x}^\text{x}\log_\text{e}(\text{ex})\text{dx}$ then $\text{g}(\pi)$ equals :
→
Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $A\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=3\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$. Then the maximum value of $\operatorname{det}(\mathrm{A})$ is........................
→
Choose the correct answer from the given four options. The corner points of the feasible region determined by the system of linear constraints are $\{(0, 0), (0, 40), (20, 40), (60, 20), (60, 0)\}$. The objective function is $Z = 4x + 3y.$ Compare the quantity in Column $A$ and Column $B.$
Column $A$
Column $B$
Maximum of $Z$
$325$
→
For $x \in R$, let the function $y(x)$ be the solution of the differential equation

$\frac{d y}{d x}+12 y=\cos \left(\frac{\pi}{12} x\right), y(0)=0 .$

Then, which of the following statements is/are $TRUE$?

→
A tangent having slope of $-\frac{4}{3}$ to the ellipse $\frac{\text{x}^2}{18}+\frac{\text{y}^2}{32}=1$ ntersects the major and minor axes at points $A$ and $B$ respectively. If $C$ is the center of the ellipse, then area of the $\triangle ABC$ is:
→
A fair $n(n > 1)$ faces die is rolled repeatedly until a number less than $n$ appears. If the mean of the number of tosses required is $\frac{ n }{9}$, then $n$ is equal to
→
If $f(x)=\frac{4 x+3}{6 x-4}, x \neq \frac{2}{3}$ and $(f \circ f)(x)=g(x)$, where $\mathrm{g}: \mathbb{R}-\left\{\frac{2}{3}\right\} \rightarrow \mathbb{R}-\left\{\frac{2}{3}\right\}$, then $(gogog) (4)$ is equal to
→