Evaluate the integral.
$\int _ { x / 3 } ^ { x / 7 } \sin t d t$
A) $- 0.500$
B) $- 1.000$
C) $- 0.400$
D) $0.250$
E) $1.000$

Question

Quizplus · Accepted Answer

Using substitution, let $u = t - \frac{x}{3}$. Then $du = dt$ and the integral becomes
\begin{align*}
\int_{x/3}^{x/7} \sin t \, dt &= \int_0^{-(4x)/21} \sin(u+(x/3)) \, du \
&= -\cos(u+(x/3))\bigg|_0^{-(4x)/21} \
&= -\cos\left(\frac{x}{3}ight) + \cos\left(\frac{4x}{21}+\frac{x}{3}ight) \
&= -\frac{2}{5}\cos\left(\frac{x}{3}ight) - \frac{3}{5}\sin\left(\frac{x}{3}ight).
\end{align*}
Therefore, the answer is $\boxed{	extbf{(C) }-0.400}$.

Evaluate the Integral ∫x/3x/7sin⁡tdt\int _ { x / 3 } ^ { x / 7 } \sin t d t∫x/3x/7​sintdt

Evaluate the Integral $\int _ { x / 3 } ^ { x / 7 } \sin t d t$