How do you integrate #int e^-xsin4x# by integration by parts method?
The answer is
We use the integration by parts
Here,
Putting it all together
Therefore,
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To integrate ( \int e^{-x} \sin(4x) ) using integration by parts:
- Select ( u = e^{-x} ) and ( dv = \sin(4x) , dx ).
- Find ( du ) and ( v ) by differentiating ( u ) and integrating ( dv ).
- Apply the integration by parts formula: ( \int u , dv = uv - \int v , du ).
- Substitute the values of ( u ), ( v ), ( du ), and ( dv ) into the formula.
- Evaluate the integral obtained in step 4.
Applying the integration by parts formula:
( u = e^{-x} ) ( du = -e^{-x} , dx )
( dv = \sin(4x) , dx ) ( v = -\frac{1}{4} \cos(4x) )
Now, applying the integration by parts formula:
( \int e^{-x} \sin(4x) , dx = -\frac{1}{4} e^{-x} \cos(4x) + \frac{1}{4} \int e^{-x} \cos(4x) , dx )
To solve ( \int e^{-x} \cos(4x) , dx ), repeat the integration by parts method.
Choose ( u = e^{-x} ) and ( dv = \cos(4x) , dx ).
( du = -e^{-x} , dx ) ( v = \frac{1}{4} \sin(4x) )
Applying the integration by parts formula again:
( \int e^{-x} \cos(4x) , dx = \frac{1}{4} e^{-x} \sin(4x) + \frac{1}{4} \int e^{-x} \sin(4x) , dx )
Substitute the result back into the original integral:
( \int e^{-x} \sin(4x) , dx = -\frac{1}{4} e^{-x} \cos(4x) + \frac{1}{4} \left( \frac{1}{4} e^{-x} \sin(4x) + \frac{1}{4} \int e^{-x} \sin(4x) , dx \right) )
Now, solve for the remaining integral:
( \frac{3}{16} \int e^{-x} \sin(4x) , dx = -\frac{1}{4} e^{-x} \cos(4x) + \frac{1}{16} e^{-x} \sin(4x) )
( \frac{13}{16} \int e^{-x} \sin(4x) , dx = -\frac{1}{4} e^{-x} \cos(4x) ) ( \int e^{-x} \sin(4x) , dx = -\frac{4}{13} e^{-x} \cos(4x) + C )
So, ( \int e^{-x} \sin(4x) , dx = -\frac{4}{13} e^{-x} \cos(4x) + 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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