# How do you integrate #e^(sinx) cosx dx#?

You can solve the integral using a u-substitution

Differentiating we get

Make the subtitution

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if you recognise the result

you can integrate this directly.

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To integrate ( e^{\sin(x)} \cos(x) , dx ), you can use integration by parts. Let ( u = e^{\sin(x)} ) and ( dv = \cos(x) , dx ). Then, ( du = e^{\sin(x)} \cos(x) , dx ) and ( v = \sin(x) ). Applying the integration by parts formula:

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

The integral on the right-hand side is of the same form as the original integral, but with ( e^{\sin(x)} \sin(x) ) instead of ( e^{\sin(x)} \cos(x) ). You can apply integration by parts again or use another method to solve it.

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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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