integral techniques.

Calculus Level 3

( e ) x d x = e x e k x 1 + b + C \large \int (-e)^x \ dx = \frac{e^xe^{kx}}{1+b} + C

Give the above, where C C is the integration constant. Find k b kb .

1 / 12 -1/12 π π π 2 -π^{2} π / 2 -π/2

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2 solutions

Chew-Seong Cheong
Dec 24, 2016

I = ( e ) x d x = ( 1 ) x e x d x By Euler’s formula: e i π = cos π + i sin π = 1 = e i π x e x d x = e ( 1 + i π ) x d x = e ( 1 + i π ) x 1 + i π + C where C is the integration constant. = e x e i π 1 + i π + C \begin{aligned} I & = \int (-e)^x \ dx \\ & = \int ({\color{#3D99F6}-1})^xe^x \ dx & \small \color{#3D99F6} \text{By Euler's formula: } e^{i\pi} = \cos \pi + i \sin \pi = -1 \\ & = \int {\color{#3D99F6}e}^{{\color{#3D99F6}i \pi}x} e^x \ dx \\ & = \int e^{(1+i\pi)x} \ dx \\ & = \frac {e^{(1+i\pi)x}}{1+i\pi} + \color{#3D99F6}C & \small \color{#3D99F6} \text{where }C \text{ is the integration constant.} \\ & = \frac {e^x e^{i\pi}}{1+i\pi} + C \end{aligned}

k = b = i π k b = 1 π 2 \implies k = b = i \pi \implies kb = \boxed{-1\pi^2}


Notation: i = 1 i = \sqrt{-1} is imaginary unit .

Subh Mandal
Dec 24, 2016

write -1 using euler form = e^(iπ) or e^(-iπ).

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