If f is an integrable function, show that: ^ a _ -a xf (x^2 )dx=0 - Vidyadip

Question

If f is an integrable function, show that:
$\int\limits^{\text{a}}_{-\text{a}}\text{xf}\big(\text{x}^2\big)\text{dx}=0$

✓

Answer

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

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" 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

Find the value of the determinant $\begin{vmatrix}4200&1201\\4205&4203\end{vmatrix}$

Write the value of $\cot^{-1}(-\text{x})$ for all $\text{x}\in\text{R}$ in terms of $\cot^{-1}\text{x}$

If A and B are square matrices of the same order, explain, why in general:
(A − B)² ≠ A² − 2AB + B²

Determine whether the following operations define a binary operation on the given set or not:

$'\odot'$ on N defined by $\text{a}\odot\text{b}=\text{a}^{\text{b}}+\text{b}^{\text{a}}$ for all $\text{a, b}\in\text{N.}$

If $\begin{bmatrix}\text{a+b}&2\\5&\text{b} \end{bmatrix}=\begin{bmatrix}6&5\\2&2 \end{bmatrix}$, then find a.

If $\begin{bmatrix}\text{xy}&4\\\text{z}+6&\text{x}+\text{y}\end{bmatrix}=\begin{bmatrix}8&\text{w}\\0&6\end{bmatrix},$ then find the values of x, y, z and w.

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\text{x}+\Big(\frac{\text{dy}}{\text{dx}}\Big)=\sqrt{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2}$

Let 'o' be a binary operation on the set Q₀ of all non-zero rational numbers defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{2}$ for all $\text{a},\text{b}\in\text{Q}_0.$
Show that 'o' is both commutative and associate.

A random variable X has the following probability distribution:

Values of X:	0	1	2	3	4	5	6	7	8
P(X)	a	3a	5a	7a	9a	11a	13a	15a	17a

Determine:

The Value of a.

Evaluate : $\int \frac{1}{\sqrt{8+3 x-x^2}} d x$.