Download App

Definite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration2 Marks
MCQ
The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\sin ^2 x}{1+2^x} d x$ is
  • A
    $\frac{\pi}{2}$
  • B
    $4 \pi$
  • ✓
    $\frac{\pi}{4}$
  • D
    $\frac{\pi}{8}$
✓

Answer

Correct option: C.
$\frac{\pi}{4}$
(C)
$\text { Let } I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\sin ^2 x}{1+2^x} d x$ ...(i)
$I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\sin ^2 x}{1+2^{-x}} d x$ ....(ii)
$\ldots\left[\because \int_{ a }^{ b } f (x) d x=\int_{ a }^{ b } f ( a + b -x) d x\right]$
Adding (i) and (ii), we get
$2 I =\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^2 x d x$
$\Rightarrow 2 I =2 \int_0^{\frac{\pi}{2}} \sin ^2 x d x$
...$\left[\begin{array}{l}\because \int_{- a }^{ a } f (x) d x=2 \int_0^{ a } f (x) d x \\ \text { if } f (x) \text { is an even function }\end{array}\right]$
$\Rightarrow I =\int_0^{\frac{\pi}{2}}\left(\frac{1-\cos 2 x}{2}\right) d x$
$=\frac{1}{2}\left[x-\frac{\sin 2 x}{2}\right]_0^{\frac{\pi}{2}}=\frac{1}{2}\left(\frac{\pi}{2}-0\right)=\frac{\pi}{4}$

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 "MCQ" from Definite Integration→Definite Integration — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

The area of the region bounded by the curve $y=x^2$ and the line $y=16$ is
→
India play two matches each with West indies and Australia. In any match the probability of india getting $0,1$ and $2$ points are $0.45, 0.05$ and $0.50$ respectively. Assuming that the outcomes are indepecdent, the probability of india getting at least 7 points is.
→
The acute angle formed between the lines joining the origin to the points of intersection of the curve $x^2+y^2-2 x-1=0$ and the line $x+y=1$ is
→
The number of solutions of the system of equations:
$2x + y − z = 7$
$x − 3y + 2z = 1$
$x + 4y − 3z = 5$
→
If $\theta=\sin ^{-1} x+\cos ^{-1} x-\tan ^{-1} x, x \geq 0$, then the smallest interval in which $\theta$ lies is given by
→
$\int_1^2 \frac{ e ^{\frac{1}{x}}}{x^2} d x=$
→
The minimum value of the function $f(x)=x \log x$ is
→
Which of the following is not correct?
→
If the function $f(x) = kx^3 - 9x^2 + 9x + 3$ is monotonically increasing in every interval, then:
→
The value of $\int\limits^1_0\tan^{-1}\Big(\frac{2\text{x}-1}{1+\text{x}-\text{x}^2}\Big)\text{ dx},$ is:
→