Download App

Definite Integrals — Maths STD 12 Science — Question

Maharashtra 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 "Solve the Following Question.(2 Marks)" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find $' k\  '$, if the equation $k x y+10 x+6 y+4=0$ represents a pair of straight lines.
→
Find $k$, such that the function
$
P (x)=\left\{\begin{array}{ll}
k\left(\frac{4}{x}\right) & ; \quad x=0,1,2,3,4, k>0 \\
0 ; & \text { otherwise, }
\end{array}\right.
$
is a probability mass function (p.m.f.)
→
Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}8&2&7\\12&3&5\\16&4&3 \end{vmatrix}$
→
Find the value of $K$, if the sum of the slopes of the lines represented by $x^2+K x y-3 y^2=0$ is twice their product.
→
For the principal values, evaluate the following:
$\sin^{-1}\Big(-\frac{\sqrt3}{2}\Big)+\text{cosec}^{-1}\Big(-\frac{2}{\sqrt3}\Big)$
→
Let $\text{f}:\Big[-\frac{\pi}{2},\frac{\pi}{2}\Big]\rightarrow\ \text{A}$ be defined by f(x)= sinx. If f is a bijection, write set A.
→
Find the value of k if the following equations represent a pair of lines : $3 x^2+10 x y+3 y^2+16 y+k=0$
→
Equation of a plane is $\bar{r} \cdot(3 \hat{i}-4 \hat{j}+12 \hat{k})=8$. Find the length of the perpendicular from the origin to the plane.
→
Evaluate the following intregals:
$\int\frac{6\text{x}-5}{\sqrt{3\text{x}^2-5\text{x}+1}}\text{ dx}$
→
Examine whether each of the following statement patterns is a tautology or a contradiction or a contingency. (p → q) ↔ (~p ∨ q)
→