Integral of x(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=x2.
Then let du=2xdx and substitute du:
∫(2u+23)du
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Integrate term-by-term:
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The integral of a constant times a function is the constant times the integral of the function:
∫2udu=2∫udu
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The integral of un is n+1un+1 when n=−1:
∫udu=2u2
So, the result is: 4u2
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The integral of a constant is the constant times the variable of integration:
∫23du=23u
The result is: 4u2+23u
Now substitute u back in:
4x4+23x2
Method #2
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Rewrite the integrand:
x(x2+3)=x3+3x
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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:
∫3xdx=3∫xdx
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The integral of xn is n+1xn+1 when n=−1:
∫xdx=2x2
So, the result is: 23x2
The result is: 4x4+23x2
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Now simplify:
4x2(x2+6)
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Add the constant of integration:
4x2(x2+6)+constant
The answer is:
4x2(x2+6)+constant
The answer (Indefinite)
[src]
/
| 4 2
| / 2 \ x 3*x
| x*\x + 3/ dx = C + -- + ----
| 4 2
/
∫x(x2+3)dx=C+4x4+23x2
The graph
Use the examples entering the upper and lower limits of integration.