How do you integrate #int y/(e^(2y))# by integration by parts method?
We have
If we say
By the integration by parts formula
Putting the constants outside of the integral
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To integrate ( \int \frac{y}{e^{2y}} ) using integration by parts, let ( u = y ) and ( dv = \frac{1}{e^{2y}} dy ). Then, differentiate ( u ) to get ( du = dy ) and integrate ( dv ) to get ( v = -\frac{1}{2e^{2y}} ).
Now, apply the integration by parts formula:
[ \int u , dv = uv - \int v , du ]
Substitute the values of ( u ), ( dv ), ( v ), and ( du ) into the formula:
[ \int \frac{y}{e^{2y}} , dy = -\frac{y}{2e^{2y}} - \int -\frac{1}{2e^{2y}} , dy ]
[ = -\frac{y}{2e^{2y}} + \frac{1}{4} \int e^{-2y} , dy ]
Now, integrate ( \int e^{-2y} ) to get:
[ = -\frac{y}{2e^{2y}} + \frac{1}{4} \left( -\frac{1}{2} e^{-2y} \right) + C ]
Where ( C ) is the constant of integration.
Therefore, the integral ( \int \frac{y}{e^{2y}} , dy ) is:
[ = -\frac{y}{2e^{2y}} - \frac{1}{8e^{2y}} + C ]
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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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