# How do you find the integral of #(cosx)(e^x)#?

Let:

We may apply integration by components:

After that, entering the IBP formula:

gives us

Subsequently, entering the IBP formula provides us with:

When we enter this outcome into [A], we obtain:

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Integrate by parts:

Now integrate again by parts:

Substituting this equality in the first one we have:

The integral now appears on both sides of the equation and we can now solve for it:

and finally:

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Recursively use integration by parts.

After 2 iterations, the negative of the integral will be on the right.

Subtract form to both sides and divide by 2.

Integration by parts:

Substitute into the formula:

Integration by parts:

Substitute into the formula:

Substitute into equation [1]:

Divide by 2:

Don't forget the constant:

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

[ \int u , dv = uv - \int v , du ]

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

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

Now, you can use integration by parts again on ( \int \sin(x) \cdot e^x , dx ), or you can recognize that it's the same integral you started with, but with ( \sin(x) ) instead of ( \cos(x) ). Therefore:

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