Integral of (5x-3/2)^9 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=5x−23.
Then let du=5dx and substitute 5du:
∫5u9du
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The integral of a constant times a function is the constant times the integral of the function:
∫u9du=5∫u9du
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The integral of un is n+1un+1 when n=−1:
∫u9du=10u10
So, the result is: 50u10
Now substitute u back in:
50(5x−23)10
Method #2
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Rewrite the integrand:
(5x−23)9=1953125x9−210546875x8+6328125x7−28859375x6+815946875x5−169568125x4+161913625x3−32492075x2+256295245x−51219683
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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:
∫1953125x9dx=1953125∫x9dx
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The integral of xn is n+1xn+1 when n=−1:
∫x9dx=10x10
So, the result is: 2390625x10
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The integral of a constant times a function is the constant times the integral of the function:
∫(−210546875x8)dx=−210546875∫x8dx
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The integral of xn is n+1xn+1 when n=−1:
∫x8dx=9x9
So, the result is: −21171875x9
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The integral of a constant times a function is the constant times the integral of the function:
∫6328125x7dx=6328125∫x7dx
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The integral of xn is n+1xn+1 when n=−1:
∫x7dx=8x8
So, the result is: 86328125x8
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The integral of a constant times a function is the constant times the integral of the function:
∫(−28859375x6)dx=−28859375∫x6dx
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The integral of xn is n+1xn+1 when n=−1:
∫x6dx=7x7
So, the result is: −21265625x7
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The integral of a constant times a function is the constant times the integral of the function:
∫815946875x5dx=815946875∫x5dx
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The integral of xn is n+1xn+1 when n=−1:
∫x5dx=6x6
So, the result is: 165315625x6
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The integral of a constant times a function is the constant times the integral of the function:
∫(−169568125x4)dx=−169568125∫x4dx
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The integral of xn is n+1xn+1 when n=−1:
∫x4dx=5x5
So, the result is: −161913625x5
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The integral of a constant times a function is the constant times the integral of the function:
∫161913625x3dx=161913625∫x3dx
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The integral of xn is n+1xn+1 when n=−1:
∫x3dx=4x4
So, the result is: 641913625x4
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The integral of a constant times a function is the constant times the integral of the function:
∫(−32492075x2)dx=−32492075∫x2dx
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The integral of xn is n+1xn+1 when n=−1:
∫x2dx=3x3
So, the result is: −32164025x3
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The integral of a constant times a function is the constant times the integral of the function:
∫256295245xdx=256295245∫xdx
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The integral of xn is n+1xn+1 when n=−1:
∫xdx=2x2
So, the result is: 512295245x2
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The integral of a constant is the constant times the variable of integration:
∫(−51219683)dx=−51219683x
The result is: 2390625x10−21171875x9+86328125x8−21265625x7+165315625x6−161913625x5+641913625x4−32164025x3+512295245x2−51219683x
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Now simplify:
51200(10x−3)10
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Add the constant of integration:
51200(10x−3)10+constant
The answer is:
51200(10x−3)10+constant
The answer (Indefinite)
[src]
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| 10
| 9 (5*x - 3/2)
| (5*x - 3/2) dx = C + -------------
| 50
/
∫(5x−23)9dx=C+50(5x−23)10
The graph
2561412081
=
2561412081
Use the examples entering the upper and lower limits of integration.