If for a continuous function f(x), _-^t (f(x)+x),dx = ^2-t^2, for all t -, then f(-3) is e…

Question

If for a continuous function $f(x)$, $\displaystyle\int_{-\pi}^t (f(x)+x)\,dx = \pi^2-t^2$, for all $t \geq -\pi$, then $f\left(-\dfrac{\pi}{3}\right)$ is equal to:

Accepted Answer

$\int_{-\pi}^t f(x)\,dx + \int_{-\pi}^t x\,dx = \pi^2-t^2$

$\int_{-\pi}^t f(x)\,dx + \dfrac{t^2}{2}-\dfrac{\pi^2}{2} = \pi^2-t^2$

$\int_{-\pi}^t f(x)\,dx = \dfrac32(\pi^2-t^2)$

Differentiating w.r.t. $t$: $f(t) = -3t$

$f\left(-\dfrac{\pi}{3}\right) = -3\left(-\dfrac{\pi}{3}\right) = \pi$

Solutions · 1