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this was discussed by prof.tkv Iyengar my teacher when he introduced complex analysis ..
dx } "{\color{black} \int e^{ax} cos(bx)dx }")
dx } "{\color{black} \int e^{ax} sin(bx)dx }")
x} dx = \frac{e^{(a+ib)x}}{a+ib}} "{\color{Yellow} \int e^{(a+ib)x} dx = \frac{e^{(a+ib)x}}{a+ib}}")
 +isin(bx))dx= \frac{e^{ax}}{a^{2}+b^{2}}(cos(bx)+isin(bx))(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)}")
 +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))} "{\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))}")
dx= \frac{e^{ax}}{a^{2}+b^{2}}(acos(bx)+bsin(bx))} "{\color{black} \int e^{ax}cos(bx)dx= \frac{e^{ax}}{a^{2}+b^{2}}(acos(bx)+bsin(bx))}")
dx= \frac{e^{ax}}{a^{2}+b^{2}}(asin(bx)-bcos(bx))} "{\color{black} \int e^{ax}sin(bx)dx= \frac{e^{ax}}{a^{2}+b^{2}}(asin(bx)-bcos(bx))}")
this was discussed by prof.tkv Iyengar my teacher when he introduced complex analysis ..
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