How do you integrate #int e^(2x)cosx# by integration by parts method?
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To integrate ( \int e^{2x} \cos(x) ) using integration by parts:
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Identify ( u ) and ( dv ): Let ( u = e^{2x} ) and ( dv = \cos(x) , dx ).
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Calculate ( du ) and ( v ): ( du = 2e^{2x} , dx ) and ( v = \sin(x) ).
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Apply the integration by parts formula: [ \int u , dv = uv - \int v , du ] Substituting the values: [ \int e^{2x} \cos(x) , dx = e^{2x} \sin(x) - \int \sin(x) \cdot 2e^{2x} , dx ]
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Integrate the remaining integral: Use integration by parts again. Let ( u = 2e^{2x} ) and ( dv = \sin(x) , dx ).
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Calculate ( du ) and ( v ): ( du = 4e^{2x} , dx ) and ( v = -\cos(x) ).
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Apply integration by parts again: [ \int 2e^{2x} \sin(x) , dx = -2e^{2x} \cos(x) - \int -\cos(x) \cdot 4e^{2x} , dx ]
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Simplify the expression: [ \int e^{2x} \cos(x) , dx = e^{2x} \sin(x) - \left( -2e^{2x} \cos(x) - \int -4e^{2x} \cos(x) , dx \right) ]
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Combine like terms and integrate the remaining integral: [ \int e^{2x} \cos(x) , dx = e^{2x} \sin(x) + 2e^{2x} \cos(x) - \int 4e^{2x} \cos(x) , dx ]
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Solve the last integral: Let ( I = \int 4e^{2x} \cos(x) , dx ). Apply integration by parts again: Let ( u = 4e^{2x} ) and ( dv = \cos(x) , dx ). Calculate ( du ) and ( v ): ( du = 8e^{2x} , dx ) and ( v = \sin(x) ). [ I = 4e^{2x} \sin(x) - \int \sin(x) \cdot 8e^{2x} , dx ] [ I = 4e^{2x} \sin(x) - 8 \int e^{2x} \sin(x) , dx ] Rearrange the equation: [ 9I = 4e^{2x} \sin(x) + C ] Where ( C ) is the constant of integration.
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Finally, solve for ( I ): [ I = \frac{4e^{2x} \sin(x) + C}{9} ]
So, the integral ( \int e^{2x} \cos(x) , dx ) is: [ e^{2x} \sin(x) + 2e^{2x} \cos(x) - \frac{4e^{2x} \sin(x) + C}{9} ]
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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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