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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