If f(x) = ∫₀ˣ tsin(t)dt, then write the value of f'(x).

Given: f(x) = ∫₀ˣ tsin(t) dt
Using Integration by parts,
f(x) = [t(-cost)]₀ˣ + [sinx]₀ˣ
=> f(x) = -xcos(x) + sinx
Now Differentiating Both sides w.r.t x:
f'(x) = -cosx + xsinx + cosx
=> f'(x) = xsinx
Alternate Method:
Given: f(x) = ∫₀ˣ tsin(t) dt
Differentiating both sides, we get
f'(x) = xsin(x) [Using Leibnitz Theorem]