How do you integrate #int e^x * cosx dx#?
I found:
I tried Integration by Parts (twice) and a little trick...!
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To integrate ( \int e^x \cdot \cos(x) , dx ), you can use integration by parts. Integration by parts formula is:
[ \int u , dv = uv - \int v , du ]
Let's choose ( u = e^x ) and ( dv = \cos(x) , dx ). Then, differentiate ( u ) to get ( du ), and integrate ( dv ) to get ( v ).
[ du = e^x , dx ] [ v = \int \cos(x) , dx = \sin(x) ]
Now, apply the integration by parts formula:
[ \int e^x \cdot \cos(x) , dx = e^x \cdot \sin(x) - \int \sin(x) \cdot e^x , dx ]
Now, we have a new integral ( \int \sin(x) \cdot e^x , dx ). To solve this, we can use integration by parts again.
Let's choose ( u = \sin(x) ) and ( dv = e^x , dx ). Then, differentiate ( u ) to get ( du ), and integrate ( dv ) to get ( v ).
[ du = \cos(x) , dx ] [ v = \int e^x , dx = e^x ]
Apply the integration by parts formula again:
[ \int e^x \cdot \cos(x) , dx = e^x \cdot \sin(x) - \left( e^x \cdot \sin(x) - \int e^x \cdot \cos(x) , dx \right) ]
Now, rearrange terms to solve for ( \int e^x \cdot \cos(x) , dx ):
[ 2\int e^x \cdot \cos(x) , dx = e^x \cdot \sin(x) + C ]
[ \int e^x \cdot \cos(x) , dx = \frac{e^x \cdot \sin(x)}{2} + C ]
So, the integral of ( \int e^x \cdot \cos(x) , dx ) is ( \frac{e^x \cdot \sin(x)}{2} + 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.
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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