Write an expression for the function, f(x), with $f^{\prime}(x)=\sin ^{2} x+\sin x$ and $f(\pi / 3)=8$ .
A) $f(x)=8+\int_{0}^{x}\left(\sin ^{2} t+\sin t\right) d t$
B) $f(x)=\int_{\pi / 3}^{x}\left(\sin ^{2} t+\sin t\right) d t-8$
C) $f(x)=8+\int_{\pi / 3}^{x}\left(\sin ^{2} t+\sin t\right) d t$
D) $f(x)=8-\frac{\pi}{3}+\int_{0}^{x}\left(\sin ^{2} t+\sin t\right) d t$

Question

Quizplus · Accepted Answer

Given $f^{\prime}(x)=\sin ^{2} x+\sin x$, to find $f(x)$, we integrate $f^{\prime}(x)$. The constant of integration is determined by the condition $f(\pi / 3)=8$. The correct expression is $f(x)=8+\int_{\pi / 3}^{x}\left(\sin ^{2} t+\sin t\right) d t$, which accounts for the initial condition by starting the integral at $\pi / 3$ and adding the initial value 8.

Write an Expression for the Function, F(x), With f′(x)=sin⁡2x+sin⁡xf^{\prime}(x)=\sin ^{2} x+\sin xf′(x)=sin2x+sinx

Write an Expression for the Function, F(x), With $f^{\prime}(x)=\sin ^{2} x+\sin x$