Koundinya-Mathematics Teachers Hub: Use of complex numbers in Integration

Twin Integrals :
To find
${\color{black} \int e^{ax} cos(bx)dx }$
and
${\color{black} \int e^{ax} sin(bx)dx }$

We begin with
${\color{black} \int e^{kx} dx = \frac{e^{kx}}{k}}$
Let k = a + ib
then,
${\color{Yellow} \int e^{(a+ib)x} dx = \frac{e^{(a+ib)x}}{a+ib}}$
${\color{black} \therefore \int e^{ax}(cos(bx) +isin(bx))dx= \frac{e^{ax}}{a^{2}+b^{2}}(cos(bx)+isin(bx))(a-ib)}$
${\color{black} \int e^{ax}(cos(bx) +isin(bx))dx= \frac{e^{ax}}{a^{2}+b^{2}}(acos(bx)+bsin(bx))+i\frac{e^{ax}}{a^{2}+b^{2}}(asin(bx)-bcos(bx))}$
Equating Real and Imaginary parts
${\color{black} \int e^{ax}cos(bx)dx= \frac{e^{ax}}{a^{2}+b^{2}}(acos(bx)+bsin(bx))}$
and
${\color{black} \int e^{ax}sin(bx)dx= \frac{e^{ax}}{a^{2}+b^{2}}(asin(bx)-bcos(bx))}$

Koundinya-Mathematics Teachers Hub

Friday, 23 December 2011

Use of complex numbers in Integration

1 comment: