# How do you find the integral of #e^x *sin x#?

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To find the integral of ( e^x \cdot \sin(x) ), you can use integration by parts. Let ( u = e^x ) and ( dv = \sin(x) , dx ). Then, ( du = e^x , dx ) and ( v = -\cos(x) ). Applying the integration by parts formula:

[ \int e^x \sin(x) , dx = -e^x \cos(x) - \int (-\cos(x)) \cdot e^x , dx ]

[ = -e^x \cos(x) + \int e^x \cos(x) , dx ]

Now, we have a new integral ( \int e^x \cos(x) , dx ), which can be solved using integration by parts again or by noticing that the integral of ( e^x \cos(x) ) is ( \frac{1}{2}e^x(\sin(x) + \cos(x)) ).

So,

[ \int e^x \sin(x) , dx = -e^x \cos(x) + \frac{1}{2}e^x(\sin(x) + \cos(x)) + C ]

Where ( C ) is the constant of integration.

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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.

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.

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.

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