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Integration by parts || Integration of e^ax sinbx || Important Question from integrals - Math Traders

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Integration by parts || Integration of e^ax sinbx || Important Question from integrals

                     Chapter: Integration by parts



In this chapter we will learn two things very clearly.

1. what is 'Integration by parts' formula
2. we will also learn how to apply 'Integration by parts formula' into our questions to find their integrals with the help of this method.

We have taken an example to which best explains this method at the end of this question we will leave few questions for you so that you can practice this method more.


Q1. Find the integration of e^ax sinbx


Sol: Let I=eaxsinbx


From integration by parts method


1stpart{2ndpart-}{(d{dx}}{1st}part{{2nd}partdx)dx}] …………………(1)



So, let 1stpart=eax        And        2ndpart=sinbx



Now put these values in eq (1) we get-



I=eaxsinbx



 I = eaxsinbx--(ddxeaxsinbxdx)dx



 I =  -1beax.cosbx+aeaxcosbxbdx



 I =  -1beax.cosbx+ab[eaxcosbxdx -(ddxeaxcosbxdx)dx]



 I =  -1beax.cosbx+ab[eaxsinbxb-a{eax.sinbxbdx}]



I =  -1beax.cosbx+ab2eaxsinbx - a2b2eax.sinbxdx



 I -1beax.cosbx+ab2eaxsinbx - a2b2.I



 I + a2b2.I=   ab2eaxsinbx- 1beax.cosbx   



  I.(a2+b2b2)eax.a.sinbx-.b.cosbxb2   



Or   I = eax.[a.sinbx-.b.cosbxa2+b2] 



Which is our final answer for the integration of eaxsinbx 



Similarly we will also find integration of eaxcosbx


e^ax Sinbx Integration by parts

                                                           👉integration of eaxcosbx


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