How do you integrate #int e^xsinx# by integration by parts method?
Integration by parts can be expressed:
and we find:
and we find:
and hence:
Integration by parts is very useful, but can end up leading you down a rabbit hole if you do not choose the parts appropriately.
So by subtracting the second from the first of these, we find:
Hence:
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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 ).

Calculate ( du ) and ( v ): ( du = e^x , dx ) and ( v = \cos(x) ).

Apply the integration by parts formula: [ \int u , dv = uv  \int v , du ]

Substitute the values: [ \int e^x \sin(x) , dx = e^x (\cos(x))  \int (\cos(x)) (e^x , dx) ]

Simplify and integrate the remaining integral: [ = e^x \cos(x) + \int e^x \cos(x) , dx ]

Repeat the integration by parts process for the new integral: Let ( u = e^x ) and ( dv = \cos(x) , dx ).

Calculate ( du ) and ( v ): ( du = e^x , dx ) and ( v = \sin(x) ).

Apply integration by parts again: [ \int e^x \cos(x) , dx = e^x \sin(x)  \int \sin(x) e^x , dx ]

Notice that the integral on the right side is the same as the original integral but with the sine and cosine functions swapped. Thus: [ \int e^x \cos(x) , dx = e^x \sin(x)  \int e^x \sin(x) , dx ]

Substitute this back into the previous equation: [ = e^x \cos(x) + e^x \sin(x)  \int e^x \sin(x) , dx ]

Add the constant of integration ( C ) to complete the solution.
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When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
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