Integral Calculation: $e^x \sin x$

First, we find the indefinite integral of $e^x \sin x$. This is a standard integral often solved using integration by parts twice.

Let $I = \int e^x \sin x \, dx$. Using integration by parts, we find:

$I = \frac{1}{2} e^x (\sin x - \cos x) + C$

Determining $A_k$

We need to evaluate the definite integral $A_k = \int_{k\pi}^{(k+1)\pi} e^x \sin x \, dx$. Using the result above:

$A_k = \left[ \frac{1}{2} e^x (\sin x - \cos x) \right]_{k\pi}^{(k+1)\pi}$

Substitute the limits:

$A_k = \frac{1}{2} \left[ e^{(k+1)\pi}(\sin((k+1)\pi) - \cos((k+1)\pi)) - e^{k\pi}(\sin(k\pi) - \cos(k\pi)) \right]$

Using $\sin(n\pi) = 0$ and $\cos(n\pi) = (-1)^n$ for integer $n$:

$A_k = \frac{1}{2} \left[ e^{(k+1)\pi}(0 - (-1)^{k+1}) - e^{k\pi}(0 - (-1)^k) \right]$ $A_k = \frac{1}{2} \left[ -e^{(k+1)\pi}(-1)^{k+1} + e^{k\pi}(-1)^k \right]$ $A_k = \frac{1}{2} \left[ e^{(k+1)\pi}(-1)^{k+2} + e^{k\pi}(-1)^k \right]$ $A_k = \frac{1}{2} \left[ e^{(k+1)\pi}(-1)^{k} + e^{k\pi}(-1)^k \right]$

Factor out common terms:

$A_k = \frac{1}{2} (-1)^k [e^{(k+1)\pi} + e^{k\pi}]$ $A_k = \frac{1}{2} (-1)^k e^{k\pi} (e^{\pi} + 1)$

Calculating Absolute Value $|A_k|$

The absolute value $|A_k|$ is:

$|A_k| = \left| \frac{1}{2} (-1)^k e^{k\pi} (e^{\pi} + 1) \right|$

Since $e^{k\pi} > 0$ and $e^{\pi} + 1 > 0$, and $|(-1)^k| = 1$, we get:

$|A_k| = \frac{1}{2} e^{k\pi} (e^{\pi} + 1)$

Summation of $|A_k|$

We need to compute the sum $\sum_{k=0}^{20} |A_k|$:

$S = \sum_{k=0}^{20} \frac{1}{2} e^{k\pi} (e^{\pi} + 1)$

Factor out the constant terms:

$S = \frac{1}{2} (e^{\pi} + 1) \sum_{k=0}^{20} (e^{\pi})^k$

The sum is a geometric series with first term $a=1$, common ratio $r = e^{\pi}$, and $n+1 = 21$ terms (from $k=0$ to $k=20$). The sum formula is $\sum_{k=0}^{n} r^k = \frac{r^{n+1}-1}{r-1}$.

$\sum_{k=0}^{20} (e^{\pi})^k = \frac{(e^{\pi})^{21} - 1}{e^{\pi} - 1} = \frac{e^{21\pi} - 1}{e^{\pi} - 1}$

Substituting this back into the expression for $S$:

$S = \frac{1}{2} (e^{\pi} + 1) \frac{e^{21\pi} - 1}{e^{\pi} - 1}$ $S = \frac{1}{2} \frac{e^{\pi}+1}{e^{\pi}-1} (e^{21\pi}-1)$