If the integration by parts was used to integrate the product of e^ax and sin(bx) and cos(bx). e.g.∫e^ax cos(bx)dx and∫e^ax sin(bx)dx.Use Euler’s Identity to evaluate the integral∫e^ax cos(bx) dx+i∫e^ax sin(x) dx =∫e^ax(cos(bx) +isin(bx)) dx =∫e^(a+ib)x dx. Remember that i is a constant!Write the number i in polar complex form and use Euler’s identity to find√i. Hint: The two roots are evenly spaced around the unit circle and√i=i^1/2.

1. To evaluate the integral ∫eaxcosbxdx+i∫eaxsinbxdx: To evaluate ∫eaxcosbxdx+i∫eaxsinbxdx, Consider, ∫eaxcosbxdx+i∫eaxsinbxdx=∫eaxcosbx+sinbxdx Applying Euler's formula eix=cosx+isinx =∫eax×eibxdx =∫ea+ibxdx =ea+ibxa+ib+C =a-ibea+ibxa2+b2+CTo represent i in polar complex form: Polar complex form of a complex number x+iy is , z=reiθ ,r=x2+y2 & θ=tan-1yx For i, r=02+12=1 & θ=tan-110=π2 Thus, i=eiπ2 To find i=i12: i12=eiπ212 =eiπ4(By using Euler'sformula) =cosπ4+isinπ4 =12+i12

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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If the integration by parts was used to integrate the product of e^ax and sin(bx) and cos(bx). e.g.∫e^ax cos(bx)dx and∫e^ax sin(bx)dx.

Use Euler’s Identity to evaluate the integral∫e^ax cos(bx) dx+i∫e^ax sin(x) dx =∫e^ax(cos(bx) +isin(bx)) dx =∫e^(a+ib)x dx. Remember that i is a constant!
Write the number i in polar complex form and use Euler’s identity to find√i. Hint: The two roots are evenly spaced around the unit circle and√i=i^1/2.

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