Evaluate the definite integral of the trig function. $\int _ { 0 } ^ { 5 \pi } ( - 3 \sin x + 4 \cos x ) d x$ Use a graphing utility to verify your results.
A) $- 7$
B) $- 5$
C) $- 6$
D) $- 8$
E) $- 4$

Question

Evaluate the definite integral of the trig function. $\int _ { 0 } ^ { 5 \pi } ( - 3 \sin x + 4 \cos x ) d x$ Use a graphing utility to verify your results. 
A) $- 7$
B) $- 5$
C) $- 6$
D) $- 8$
E) $- 4$

Quizplus · Accepted Answer

Using integration by parts, we get:

$\int ( - 3 \sin x + 4 \cos x ) d x = - 3 \cos x - 4 \sin x + C$

Evaluating the definite integral from $0$ to $5\pi$:

$\int _ { 0 } ^ { 5 \pi } ( - 3 \sin x + 4 \cos x ) d x = [- 3 \cos x - 4 \sin x ]_0^{5\pi} = -3\cos(5\pi) -4\sin(5\pi) + 3\cos(0) +4\sin(0) = -6$

Using a graphing utility, we can plot the function and find the area under the curve from $0$ to $5\pi$, which is approximately $-6$.

Evaluate the Definite Integral of the Trig Function ∫05π(−3sin⁡x+4cos⁡x)dx\int _ { 0 } ^ { 5 \pi } ( - 3 \sin x + 4 \cos x ) d x∫05π​(−3sinx+4cosx)dx

Evaluate the Definite Integral of the Trig Function $\int _ { 0 } ^ { 5 \pi } ( - 3 \sin x + 4 \cos x ) d x$