Determination of Function
General
Grade 12
Question:
$\int_0^x f(t)dt = x \cos \pi x, x > 0$ find $f(4)$
Step-by-Step Solution
Key Concept: General
Differentiating both sides with respect to $x$ using Leibniz rule:<br>$\frac{d}{dx} \int_0^x f(t)dt = \frac{d}{dx} (x \cos \pi x)$<br>$f(x) = \cos \pi x - \pi x \sin \pi x$<br>For $x=4$, $f(4) = \cos 4\pi - 4\pi \sin 4\pi = 1 - 4\pi(0) = 1$
Correct Answer: 1