Provide an appropriate response.
-Find $f ( \pi )$ if $\int _ { 0 } ^ { x } f ( t ) d t = x \sin 8 x$.
A) 8
B) $8 \pi$
C) 0
D) $\pi$

Question

Quizplus · Accepted Answer

We can use the Fundamental Theorem of Calculus to find $f(x)$ by taking the derivative of both sides of the given equation:
$$\frac{d}{dx} \left( \int_0^x f(t) dt \right) = \frac{d}{dx} (x \sin 8x)$$
$$f(x) = \frac{d}{dx} (x \sin 8x) = \sin 8x + 8x \cos 8x$$
So, $f(\pi) = \sin 8\pi + 8\pi \cos 8\pi = 0 + 8\pi(-1) = \boxed{-8\pi}$.

Provide an Appropriate Response f(π)f ( \pi )f(π) If ∫0xf(t)dt=xsin⁡8x\int _ { 0 } ^ { x } f ( t ) d t = x \sin 8 x∫0x​f(t)dt=xsin8x

Provide an Appropriate Response $f ( \pi )$ If $\int _ { 0 } ^ { x } f ( t ) d t = x \sin 8 x$