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Q.
Let $f(x)$ be a continuous function in R such that $f(x) + f(y) = f(x + y)$. If $\int _0^3 f(x)dx=$K, then $\int_{-3}^3f(x) dx $ is equal to
Integrals
Solution:
Since $f \left(x\right)+f \left(y\right)=f \left(x+y\right) ...\left(1\right)$
$\therefore f \left(x\right)+f \left(-x\right)=f \left(0\right) \ldots\left(2\right)$
[By putting $y = - x$ in $(1)]$
Again $f\left(0\right)+f\left(0\right)=f \left(0\right)$
[By putting $x = y =0$ in $(1)$]
$\Rightarrow f \left(0\right)=0 \therefore \left(2\right)$ gives
$f\left(x\right)+f\left(-x\right)=0 \therefore f \left(-x\right)=-f\left(x\right)$
$\therefore f\left(x\right)$ is an odd function.
$\therefore \int \limits_{-3}^{3} f \left(x\right)dx=0$