Use of complex numbers in calculus.

Hi, you might have come across: $\displaystyle \int e^{ax}\cos bx$ dx,

How do you solve it? You might use integration by parts, and also complex numbers, and I find use of complex numbers interesting!

Say $A = \displaystyle \int e^{ax}\cos bx$ dx,

and $B = \displaystyle \int e^{ax}\sin bx$ dx

Hence, $A + iB = \displaystyle \int e^{ax} (\cos bx + i \sin bx)$ dx = $\displaystyle \int e^{ax} (e^{i bx})$ dx

= $\displaystyle \int e^{(a+ib)x}$ dx

= $\displaystyle \frac{e^{(a+ib)x}}{a+ib}$

= $\displaystyle \frac{e^{(a+ib)x}(a - ib)}{a^2 + b^2}$

$\Rightarrow A + iB = z = \displaystyle \frac{e^{ax} (\cos bx + i \sin bx)(a - ib)}{a^2+b^2}$

Clearly,

$A = \text{Re}(z) = \displaystyle \frac{e^{ax}}{a^2+b^2} (a \cos bx + b \sin bx)$

$B = \text{Im}(z) = \displaystyle \frac{e^{ax}}{a^2+b^2} (a \sin bx - b \cos bx)$

You can try to find this definite integral:

Problem: $\displaystyle \int_{0}^{\pi} e^{(\cos x)} \cos(\sin x)$ dx

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