Integral of x(e^(2x)) dx
The solution
Detail solution
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=x and let dv(x)=e2x.
Then du(x)=1.
To find v(x):
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There are multiple ways to do this integral.
Method #1
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Let u=2x.
Then let du=2dx and substitute 2du:
∫4eudu
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The integral of a constant times a function is the constant times the integral of the function:
∫2eudu=2∫eudu
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The integral of the exponential function is itself.
∫eudu=eu
So, the result is: 2eu
Now substitute u back in:
2e2x
Method #2
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Let u=e2x.
Then let du=2e2xdx and substitute 2du:
∫41du
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The integral of a constant times a function is the constant times the integral of the function:
∫21du=2∫1du
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The integral of a constant is the constant times the variable of integration:
∫1du=u
So, the result is: 2u
Now substitute u back in:
2e2x
Now evaluate the sub-integral.
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The integral of a constant times a function is the constant times the integral of the function:
∫2e2xdx=2∫e2xdx
-
Let u=2x.
Then let du=2dx and substitute 2du:
∫4eudu
-
The integral of a constant times a function is the constant times the integral of the function:
∫2eudu=2∫eudu
-
The integral of the exponential function is itself.
∫eudu=eu
So, the result is: 2eu
Now substitute u back in:
2e2x
So, the result is: 4e2x
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Now simplify:
4(2x−1)e2x
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Add the constant of integration:
4(2x−1)e2x+constant
The answer is:
4(2x−1)e2x+constant
The answer (Indefinite)
[src]
/
| 2*x 2*x
| 2*x e x*e
| x*e dx = C - ---- + ------
| 4 2
/
4(2x−1)e2x
The graph
4e2+41
=
41+4e2
Use the examples entering the upper and lower limits of integration.