Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS1 Mark
MCQ
$\int\limits^{2\text{a}}_0\text{f}(\text{x})\text{dx}$ is equal to:
  • A
    $2\int\limits^{\text{a}}_0\text{f(x)}\text{dx}$
  • B
    $0$
  • ✓
    $\int\limits^{\text{a}}_0\text{f}(\text{x})\text{dx}+\int\limits^{\text{a}}_0\text{f}(2\text{a}-\text{x})\text{dx}$
  • D
    $\int\limits^{\text{a}}_0\text{f}(\text{x})\text{dx}+\int\limits^{2\text{a}}_0\text{f}(2\text{a}-\text{x})\text{dx}$
✓

Answer

Correct option: C.
$\int\limits^{\text{a}}_0\text{f}(\text{x})\text{dx}+\int\limits^{\text{a}}_0\text{f}(2\text{a}-\text{x})\text{dx}$
$\int\limits^\text{a}_0\text{f}(\text{x})\text{dx}+\int\limits^\text{a}_0\text{f}(2\text{a}-\text{x})\text{dx}$

According to the additivity property of integrals,

$\int\limits^\text{b}_\text{a}\text{f}(\text{x})\text{dx}=\int\limits^\text{c}_\text{a}\text{f}(\text{x})+\int\limits^\text{b}_\text{c}\text{f}(\text{x})\text{dx},$ where a < c < b

Using this property,

$\int\limits^{2\text{a}}_0\text{f}(\text{x})\text{dx}=\int\limits^\text{a}_0\text{f}(\text{x})\text{dx}+\int\limits^{2\text{a}}_0\text{f}(\text{x})\text{dx}\ ....(\text{i})$

Now, consider the integral, $\int\limits^{2\text{a}}_0\text{f}(\text{x})\text{dx}$

Let x = 2a - t Then dx = d (2a - t), dx = - dt

Also, x = a, t = a and x = 2a, t = 0

Therefore, $\int\limits^{2\text{a}}_\text{a}\text{f}(\text{x})\text{dx}=-\int\limits^0_\text{a}\text{f}(2\text{a}-\text{t})\text{dt}$

$\Rightarrow \int\limits^{2\text{a}}_\text{a}\text{f}(\text{x})\text{dx}=\int\limits^\text{a}_0\text{f}(2\text{a}-\text{t})\text{dt}$

$\Rightarrow \int\limits^{2\text{a}}_\text{a}\text{f}(\text{x})\text{dx}=\int\limits^\text{a}_0\text{f}(2\text{a}-\text{x})\text{dx}$

Substituting this in equation (i) we get,

$\int\limits^{2\text{a}}_0\text{f}(\text{x})\text{dx}=\int\limits^\text{a}_0\text{f}(\text{x})\text{dx}+\int\limits^\text{a}_0\text{f}(2\text{a}-\text{x})\text{dx}$

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

$A = \left[ {\begin{array}{*{20}{c}}5&{ - 3}\\2&4\end{array}} \right]$and $B = \left[ {\begin{array}{*{20}{c}}6&{ - 4}\\3&6\end{array}} \right],$ then $A - B = $
→
The value of $‘a’$  in order that $f(x) = \sqrt 3 $ $\sin x - \cos x - 2ax + b$ decreases for all real values of  $x$, is given by
→
If ${I_n} = \int\limits_{ - n}^n {{{\tan }^2}\left\{ x \right\}dx} $ then (where {.} denotes fractional part function and $n \in  N$ )
→
Let a curve $y=y(x)$ be given by the solution of the differential equation

$\cos \left(\frac{1}{2} \cos ^{-1}\left(e^{-x}\right)\right) d x=\sqrt{e^{2 x}-1} \,d y$

If it intersects $y$-axis at $y=-1$, and the intersection point of the curve with $x$-axis is $(\alpha, 0)$ the $\mathrm{e}^{\alpha}$ is equal to $.....$

→
Evaluate$:\ \int\frac{1-\cos\text{x}}{cos\text{x}(1+cos\text{x})}\text{dx}.$
→
If $y=\left(1+\frac{1}{x}\right)^x$, then $\frac{d y}{d x}=$
→
Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]$ and $\mathrm{B}=\left[\mathrm{b}_{\mathrm{ij}}\right]$ be two $3 \times 3$ real matrices such that $b_{i j}=(3)^{(i+j-2)} a_{j i},$ where $\mathrm{i}, \mathrm{j}=1,2,3 .$ If the determinant of $\mathrm{B}$ is $81,$ then the determinant of $A$ is
→
The solution of the differential equation $\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}+\text{x}\ \tan\frac{\text{y}}{\text{x}}$ is:
→
A speaks truth in 75% cases and B seaks truth in 80% cases. Probability that they contradict each other in a statement, is
→
Let $f$ be differentiable at $x = 0$ and $f ‘ (0) = 1$. Then $\mathop {Lim}\limits_{h \to 0} \,\,\frac{{{\text{f}}(h) - {\text{f}}( - 2h)}}{h}$=
→