Cos ( x i )

Calculus Level 3

If $\displaystyle I=\int_0^{\frac{\pi}{2}}\cos\left(xi\right)\ dx$ , enter $| I |$ .

Notation: $i=\sqrt{-1}$ denotes the imaginary unit .

The answer is 2.30129890231.

1 solution

Kellen Atkins
Sep 4, 2018

$e^{xi}=\cos\left(x\right)+\sin\left(x\right)i\\e^{-xi}=\cos\left(x\right)-\sin\left(x\right)i\\e^{xi}+e^{-xi}=2\cos\left(x\right)\\\frac{e^{xi}+e^{-xi}}{2}=\cos\left(x\right)\\ \int_0^{\frac{\pi}{2}}\cos\left(xi\right)\ dx=\int_0^{\frac{\pi}{2}}\frac{e^{-x}+e^x}{2}dx=\int_0^{\frac{\pi}{2}}\cosh\left(x\right)dx=\sinh\left(\frac{\pi}{2}\right)-\sinh\left(0\right)=\boxed{2.3012...}$

How did you changed the last part from e^x to hyperbolic cosine? Why not directly solve that integral of e^x?

Department 8 - 2 years, 9 months ago

It's the same thing it doesn't really matter.

kellen atkins - 2 years, 9 months ago

Cos ( x i )

The answer is 2.30129890231.

1 solution

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