Download App

Definite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals2 Marks
Question
If f is an integrable function, show that:
$\int\limits^{\text{a}}_{-\text{a}}\text{f}\big(\text{x}^2\big)\text{dx}=2\int\limits^\text{a}_0\text{f}\big(\text{x}^2\big)\text{dx}$
✓

Answer

$\text{I}=\int\limits^{\text{a}}_{-\text{a}}\text{f}\big(\text{x}^2\big)\text{dx}$
Here, $\text{g(x)}=\text{f}\big(\text{x}^2\big)$
$\Rightarrow\text{g}(-\text{x})=\text{f}(-\text{x})^2=\text{f}(\text{x})^2=\text{g(x)}\text{ i.e., g(x) is even}$
Therefore,
$\text{I}=2\int\limits^{\text{a}}_{0}\text{f}(\text{x}^2)\text{dx}$ $\bigg[\text{Using}\text{I}=\int\limits^{\text{a}}_{-\text{a}}\text{g}(\text{x})\text{dx}=\int\limits^{\text{a}}_{0}\text{f}(\text{x})\text{dx}\text{ when g(x) is even}\bigg]$

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 "2 Marks Questions" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following:
$\sin^{-1}\Big\{\Big(\sin-\frac{17\pi}{8}\Big)\Big\}$
→
Write the point where $\text{f}(\text{x})=\text{x}\log_{\text{e}}\text{x}$ attains minimum value.
→
If $\cot ^{-1}(3 x+5)>\frac{\pi}{4}$, then find the range of the values of $x$
→
Evaluate the following integrals:$\int\frac{\text{e}^\text{x}}{\sqrt{16-\text{e}^{2\text{x}}}}\text{ dx}$
→
If x and y are connected parametrically by the equations given in Exercise without eliminating the parameter, Find $\frac{\text{dy}}{\text{dx}}.$
$\text{x}=\text{a}\cos\theta,\text{y}=\text{b}\cos\theta$
→
Evaluate the following integrals:
$\int\sin^5\text{x}\cos\text{x dx}$
→
Define direction cosines of a direction line.
→
Evaluate the following integrals:
$\int\cot^{-1}\Big(\frac{\sin2\text{x}}{1-\cos2\text{x}}\Big)\text{dx}$
→
Which one the following relations on A = {1, 2, 3} is a function?
f = {(1, 3), (2, 3), (3, 2)}, g = {(1, 2), (1, 3), (3, 1)}
→
In a parliament election, a political party hired a public relations firm to promote its candidates in three ways - telephone, house calls and letters. The cost per contact (in paisa) is given in matrix A as
$\text{A}=\begin{bmatrix}140&\text{Telephone}\\200&\text{House calls}\\150&\text{Letters}\end{bmatrix}$
The number of contacts of each type made in two cities X and Yis given in the matrix B as
$\begin{matrix}\text{Telephone}&\text{House calls}&\text{Letters}\end{matrix}\\\text{B}=\begin{bmatrix}1000&500&5000\\3000&1000&10000\end{bmatrix}\begin{matrix}\text{City X}\\\text{City Y}\end{matrix}$
Find the total amount spent by the party in the two cities.
What should one consider before casting his/ her vote - party's promotional activity of their social activities?
→