Integral of (x-2)^3 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−2.
Then let du=dx and substitute du:
∫u3du
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The integral of un is n+1un+1 when n=−1:
∫u3du=4u4
Now substitute u back in:
4(x−2)4
Method #2
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Rewrite the integrand:
(x−2)3=x3−6x2+12x−8
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Integrate term-by-term:
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The integral of xn is n+1xn+1 when n=−1:
∫x3dx=4x4
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The integral of a constant times a function is the constant times the integral of the function:
∫(−6x2)dx=−6∫x2dx
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The integral of xn is n+1xn+1 when n=−1:
∫x2dx=3x3
So, the result is: −2x3
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The integral of a constant times a function is the constant times the integral of the function:
∫12xdx=12∫xdx
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The integral of xn is n+1xn+1 when n=−1:
∫xdx=2x2
So, the result is: 6x2
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The integral of a constant is the constant times the variable of integration:
∫(−8)dx=−8x
The result is: 4x4−2x3+6x2−8x
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Now simplify:
4(x−2)4
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Add the constant of integration:
4(x−2)4+constant
The answer is:
4(x−2)4+constant
The answer (Indefinite)
[src]
/
| 4
| 3 (x - 2)
| (x - 2) dx = C + --------
| 4
/
∫(x−2)3dx=C+4(x−2)4
The graph
Use the examples entering the upper and lower limits of integration.