# Let f be a continuous function: a) Find f(4) if ∫_0^(x^2) f(t) dt = x sin πx for all x. b) Find f(4) if ∫_0^ f(x)t^2 dt = x sin πx for all x?

May 30, 2016

a) $f \left(4\right) = \frac{\pi}{2}$; b) $f \left(4\right) = 0$

#### Explanation:

a) Differentiate both sides.

Through the Second Fundamental Theorem of Calculus on the left-hand side and the product and chain rules on the right-hand side, we see that differentiation reveals that:

$f \left({x}^{2}\right) \cdot 2 x = \sin \left(\pi x\right) + \pi x \cos \left(\pi x\right)$

Letting $x = 2$ shows that

$f \left(4\right) \cdot 4 = \sin \left(2 \pi\right) + 2 \pi \cos \left(2 \pi\right)$

$f \left(4\right) \cdot 4 = 0 + 2 \pi \cdot 1$

$f \left(4\right) = \frac{\pi}{2}$

b) Integrate the interior term.

${\int}_{0}^{f} \left(x\right) {t}^{2} \mathrm{dt} = x \sin \left(\pi x\right)$

${\left[{t}^{3} / 3\right]}_{0}^{f} \left(x\right) = x \sin \left(\pi x\right)$

Evaluate.

${\left(f \left(x\right)\right)}^{3} / 3 - {0}^{3} / 3 = x \sin \left(\pi x\right)$

${\left(f \left(x\right)\right)}^{3} / 3 = x \sin \left(\pi x\right)$

${\left(f \left(x\right)\right)}^{3} = 3 x \sin \left(\pi x\right)$

Let $x = 4$.

${\left(f \left(4\right)\right)}^{3} = 3 \left(4\right) \sin \left(4 \pi\right)$

${\left(f \left(4\right)\right)}^{3} = 12 \cdot 0$

$f \left(4\right) = 0$