Chapter: Integration by parts
In this chapter we will learn two things very clearly.1. what is 'Integration by parts' formula2. 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 = eax∫sinbx--∫(ddxeax∫sinbxdx)dx
I = -1beax.cosbx+∫aeaxcosbxbdx
I = -1beax.cosbx+ab[eax∫cosbxdx -∫(ddxeax∫cosbxdx)dx]
I = -1beax.cosbx+ab[eaxsinbxb-a∫{eax.sinbxbdx}]
I = -1beax.cosbx+ab2eaxsinbx - a2b2∫eax.sinbxdx
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
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