Integral of -4*x*exp(-2*x) dx
The solution
Detail solution
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=−4x and let dv(x)=e−2x.
Then du(x)=−4.
To find v(x):
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Let u=−2x.
Then let du=−2dx and substitute −2du:
∫(−2eu)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: −2eu
Now substitute u back in:
−2e−2x
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:
∫2e−2xdx=2∫e−2xdx
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Let u=−2x.
Then let du=−2dx and substitute −2du:
∫(−2eu)du
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The integral of a constant times a function is the constant times the integral of the function:
-
The integral of the exponential function is itself.
∫eudu=eu
So, the result is: −2eu
Now substitute u back in:
−2e−2x
So, the result is: −e−2x
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Now simplify:
(2x+1)e−2x
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Add the constant of integration:
(2x+1)e−2x+constant
The answer is:
(2x+1)e−2x+constant
The answer (Indefinite)
[src]
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| -2*x -2*x -2*x
| -4*x*e dx = C + 2*x*e + e
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∫−4xe−2xdx=C+2xe−2x+e−2x
The graph
−1+e23
=
−1+e23
Use the examples entering the upper and lower limits of integration.