# How do you integrate #int e^(5x)cos3x#?

You can integrate by parts twice and algebraically solve for the integral to get

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To integrate ( \int e^{5x}\cos(3x) ), you can use integration by parts. Let ( u = e^{5x} ) and ( dv = \cos(3x) dx ). Then, ( du = 5e^{5x} dx ) and ( v = \frac{1}{3} \sin(3x) ). Applying the integration by parts formula ( \int u , dv = uv - \int v , du ), you'll get:

[ \int e^{5x}\cos(3x) , dx = \frac{1}{3} e^{5x} \sin(3x) - \frac{5}{3} \int e^{5x} \sin(3x) , dx ]

This integral on the right side can be further solved by another integration by parts or by using other techniques like integration of trigonometric functions with exponential functions.

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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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