int_{-1}^1 (a x^3 + b x) d x = 0 for | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $I = \int_{-1}^1 (a x^3 + b x) d x$.The integrand $f(x) = a x^3 + b x$ is an odd function because $f(-x) = a(-x)^3 + b(-x) = -(a x^3 + b x) =

Vedclass · Accepted Answer

any value of $a$ and $b$

Vedclass · Answer

(A) Let $I = \int_{-1}^1 (a x^3 + b x) d x$.The integrand $f(x) = a x^3 + b x$ is an odd function because $f(-x) = a(-x)^3 + b(-x) = -(a x^3 + b x) = -f(x)$.According to the property of definite integrals,if $f(x)$ is an odd function,then $\int_{-a}^a f(x) d x = 0$.Alternatively,evaluating the integral:$\int_{-1}^1 (a x^3 + b x) d x = \left[ a \frac{x^4}{4} + b \frac{x^2}{2} \right]_{-1}^1$$= \left( \frac{a(1)^4}{4} + \frac{b(1)^2}{2} \right) - \left( \frac{a(-1)^4}{4} + \frac{b(-1)^2}{2} \right)$$= \left( \frac{a}{4} + \frac{b}{2} \right) - \left( \frac{a}{4} + \frac{b}{2} \right) = 0$.This result holds true for any real values of $a$ and $b$.

$\int_{-1}^1 (a x^3 + b x) d x = 0$ for

Similar Questions

During implantation,the blastocyst becomes embedded in which layer of the uterus?

Two wires are made of the same material and have the same volume. Wire $1$ has cross-sectional area $A$ and wire $2$ has cross-sectional area $3A$. If the length of wire $1$ increases by $\Delta x$ on applying force $F$,how much force is needed to stretch wire $2$ by the same amount $\Delta x$?

An inductive circuit contains a resistance of $10 \, \Omega$ and an inductance of $2.0 \, H$. If an $AC$ voltage of $120 \, V$ and frequency of $60 \, Hz$ is applied to this circuit,the current in the circuit would be nearly......$A$.

Which of the following is not a conjugate acid-base pair?

Vedclass Test Series

Exam Paper Generator

Online Exam Module