How do you integrate #int e^x sin x dx # using integration by parts?

Answer 1

#inte^xsin(x)dx = (e^xsin(x) - e^xcos(x))/2 + c#

Say #dv = e^x# so #v = e^x#, #u = sin(x)# so #du = cos(x)#

#inte^xsin(x)dx = e^xsin(x) - inte^(x)cos(x)dx#

Say #dv = e^x# so #v = e^x#, #u = cos(x)# so #du = -sin(x)#

#inte^xsin(x)dx = e^xsin(x) - (e^xcos(x) +inte^xsin(x)dx)# #inte^xsin(x)dx = e^xsin(x) - e^xcos(x) -inte^xsin(x)dx# #2inte^xsin(x)dx = e^xsin(x) - e^xcos(x)# #inte^xsin(x)dx = (e^xsin(x) - e^xcos(x))/2 + c#

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Answer 2

Scarlett Allison

To integrate ( \int e^x \sin(x) , dx ) using integration by parts:

Choose ( u ) and ( dv ): Let ( u = e^x ) and ( dv = \sin(x) , dx ).
Find ( du ) and ( v ): Differentiate ( u ) to find ( du ), and integrate ( dv ) to find ( v ): ( du = e^x , dx ) and ( v = -\cos(x) ).
Apply the integration by parts formula: ( \int u , dv = uv - \int v , du ).
Substitute ( u ), ( dv ), ( du ), and ( v ) into the integration by parts formula.
Perform the necessary calculations to evaluate the integral.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.