Home › JEE Advanced › Mathematics › Definite Integrals › If $f$ is **odd** on $[-a, a]$, then $\displayst…
If $f$ is **odd** on $[-a, a]$, then $\displaystyle\int_{-a}^{a} f(x)\, dx$ equals:
A$0$
B$2\int_0^a f(x)dx$
C$a$
DCannot be determined
Answer & Solution
Correct answer: A. $0$
By symmetry, odd-function integrals over symmetric intervals vanish. (For even $f$: $2\int_0^a f$.)
Related questions
$\displaystyle\int_0^{\infty} \dfrac{\ln x}{1 + x^2}\, dx = ?$$\displaystyle\int_0^1 \dfrac{x - 1}{\ln x}\, dx = ?$ (Frullani-type)Given $\displaystyle\int_0^{\pi/2} \dfrac{ in^n x}{ in^n x + \cos^n x}\, dx$, the value isIf $\displaystyle f(x) = \int_0^x t in t\, dt$, then $f''(\pi) = ?$$\displaystyle\int_0^{\pi/2} \dfrac{dx}{1 + \tan^3 x} = ?$Evaluate $\displaystyle\lim_{n\to\infty} \dfrac{1^k + 2^k + \cdots + n^k}{n^{k+1}}$ for $k$\displaystyle\int_0^\pi \dfrac{x}{1 + in x}\, dx = ?$$\displaystyle\int_0^{\pi/4} \tan^4 x\, dx = ?$