Download App

STD 12 - 7.2 definite integral — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 7.2 definite integral1 Mark
MCQ
The integral $\int_{0}^{\frac{\pi}{2}} \frac{1}{3+2 \sin x+\cos x} d x$ is equal to.
  • A
    $\tan ^{-1}(2)$
  • ✓
    $\tan ^{-1}(2)-\frac{\pi}{4}$
  • C
    $\frac{1}{2} \tan ^{-1}(2)-\frac{\pi}{8}$
  • D
    $\frac{1}{2}$
✓

Answer

Correct option: B.
$\tan ^{-1}(2)-\frac{\pi}{4}$
b
$I=\int_{0}^{\frac{\pi}{2}} \frac{d x}{3+2 \sin x+\cos x}=\int_{0}^{\frac{\pi}{2}} \frac{\sec ^{2} \frac{x}{2} \cdot d x}{2 \tan ^{2} \frac{x}{2}+4 \tan \frac{x}{2}+4}$

Put $\tan \frac{x}{2}=t$, so

$I=\int_{0}^{1} \frac{d t}{(t+1)^{2}+1}=\left.\tan ^{-1}(x+1)\right|_{0} ^{1}=\tan ^{-1} 2-\frac{\pi}{4}$

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 - 7.2 definite integral→STD 12 - 7.2 definite integral — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

A four$-$digit number is formed by using the digits $1, 2, 4, 8$ and $9$ without repitition. If one number is selected from those numbers, then what is the probability that it will be an odd number?
→
Let $X_n=\{1,2,3, \ldots, n\}$ and let a subset $A$ of $X_n$ be chosen so that every pair of elements of $A$ differ by at least 3. (For example, if $n=5, A$ can be $\phi,\{2\}$ or $\{1,5\}$ among others). When $n=10$, let the probability that $1 \in A$ be $p$ and let the probability that $2 \in A$ be Then,
→
Let $f(\mathrm{x})=\left(\sin \left(\tan ^{-1} \mathrm{x}\right)+\sin \left(\cot ^{-1} \mathrm{x}\right)\right)^{2}-1,|\mathrm{x}|>1$ If $\frac{d y}{d x}=\frac{1}{2} \frac{d}{d x}\left(\sin ^{-1}(f(x))\right) $ and $ y(\sqrt{3})=\frac{\pi}{6}$ then $y(-\sqrt{3})$ is equal to
→
The number of points at which the function $f(x) = |x - 0.5| + |x - 1| + \tan x$ does not have a derivative in the interval $(0, 2),$ is
→
If $\frac{d y}{d x}=\frac{2^{x+y}-2^{x}}{2^{y}}, y(0)=1$, then $y(1)$ is equal to :
→
Shortest distance between the lines $\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}$ and $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3}$ is
→
If $|A|=|k A|$, where $A$ is a square matrix of order 2 , then sum of all possible values of $k$ is
→
The circumference of a circle is measured as $28\ cm$ with an error of $0.01\ cm$. The percentage error in the area is :
→
Let $A=\{-4,-3,-2,0,1,3,4\}$ and $R =\{( a , b ) \in A$ $\times A : b =| a |$ or $\left.b ^2= a +1\right\}$ be a relation on $A$. Then the minimum number of elements, that must be added to the relation $R$ so that it becomes reflexive and symmetric, is $........$.
→
Let $f$ : $A \to B$ be a function defined as $f(x)\, = \frac{{x - 1}}{{x - 2}}$ , where $A\, = R - \{2\}$ and $B\, = R - \{1\}$ . Then $f$ is
→