13. -/6.25 points details my notes this question has several parts that must be completed sequentially. if…

13. -/6.25 points details my notes this question has several parts that must be completed sequentially. if you skip a part of the questic tutorial exercise evaluate the integral. ∫e^7θ sin(8θ) dθ step 1 we will begin by letting u = sin(8θ) and dv = e^7θ dθ. then du = dθ and v = submit skip (you cannot come back) need help? read it

13. -/6.25 points details my notes this question has several parts that must be completed sequentially. if you skip a part of the questic tutorial exercise evaluate the integral. ∫e^7θ sin(8θ) dθ step 1 we will begin by letting u = sin(8θ) and dv = e^7θ dθ. then du = dθ and v = submit skip (you cannot come back) need help? read it

Answer

Explanation:

Step1: Find $du$

Differentiate $u = \sin(8\theta)$ with respect to $\theta$. Using the chain - rule, if $y=\sin(u)$ and $u = 8\theta$, then $\frac{dy}{d\theta}=\frac{dy}{du}\cdot\frac{du}{d\theta}$. Since $\frac{d}{du}\sin(u)=\cos(u)$ and $\frac{d}{d\theta}(8\theta)=8$, we have $du = 8\cos(8\theta)d\theta$.

Step2: Find $v$

Integrate $dv=e^{7\theta}d\theta$ with respect to $\theta$. Using the formula $\int e^{ax}dx=\frac{1}{a}e^{ax}+C$ (where $a = 7$), we get $v=\frac{1}{7}e^{7\theta}$.

Answer:

$du = 8\cos(8\theta)d\theta$, $v=\frac{1}{7}e^{7\theta}$