Integral of (x+1)^5 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+1.
Then let du=dx and substitute du:
∫u5du
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The integral of un is n+1un+1 when n=−1:
∫u5du=6u6
Now substitute u back in:
6(x+1)6
Method #2
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Rewrite the integrand:
(x+1)5=x5+5x4+10x3+10x2+5x+1
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Integrate term-by-term:
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The integral of xn is n+1xn+1 when n=−1:
∫x5dx=6x6
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The integral of a constant times a function is the constant times the integral of the function:
∫5x4dx=5∫x4dx
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The integral of xn is n+1xn+1 when n=−1:
∫x4dx=5x5
So, the result is: x5
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The integral of a constant times a function is the constant times the integral of the function:
∫10x3dx=10∫x3dx
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The integral of xn is n+1xn+1 when n=−1:
∫x3dx=4x4
So, the result is: 25x4
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The integral of a constant times a function is the constant times the integral of the function:
∫10x2dx=10∫x2dx
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The integral of xn is n+1xn+1 when n=−1:
∫x2dx=3x3
So, the result is: 310x3
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The integral of a constant times a function is the constant times the integral of the function:
∫5xdx=5∫xdx
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The integral of xn is n+1xn+1 when n=−1:
∫xdx=2x2
So, the result is: 25x2
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The integral of a constant is the constant times the variable of integration:
∫1dx=x
The result is: 6x6+x5+25x4+310x3+25x2+x
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Now simplify:
6(x+1)6
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Add the constant of integration:
6(x+1)6+constant
The answer is:
6(x+1)6+constant
The answer (Indefinite)
[src]
/
| 6
| 5 (x + 1)
| (x + 1) dx = C + --------
| 6
/
∫(x+1)5dx=C+6(x+1)6
The graph
Use the examples entering the upper and lower limits of integration.