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