Integral of (2x-3)^7 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=2x−3.
Then let du=2dx and substitute 2du:
∫4u7du
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The integral of a constant times a function is the constant times the integral of the function:
∫2u7du=2∫u7du
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The integral of un is n+1un+1 when n=−1:
∫u7du=8u8
So, the result is: 16u8
Now substitute u back in:
16(2x−3)8
Method #2
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Rewrite the integrand:
(2x−3)7=128x7−1344x6+6048x5−15120x4+22680x3−20412x2+10206x−2187
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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:
∫128x7dx=128∫x7dx
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The integral of xn is n+1xn+1 when n=−1:
∫x7dx=8x8
So, the result is: 16x8
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The integral of a constant times a function is the constant times the integral of the function:
∫(−1344x6)dx=−1344∫x6dx
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The integral of xn is n+1xn+1 when n=−1:
∫x6dx=7x7
So, the result is: −192x7
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The integral of a constant times a function is the constant times the integral of the function:
∫6048x5dx=6048∫x5dx
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The integral of xn is n+1xn+1 when n=−1:
∫x5dx=6x6
So, the result is: 1008x6
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The integral of a constant times a function is the constant times the integral of the function:
∫(−15120x4)dx=−15120∫x4dx
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The integral of xn is n+1xn+1 when n=−1:
∫x4dx=5x5
So, the result is: −3024x5
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The integral of a constant times a function is the constant times the integral of the function:
∫22680x3dx=22680∫x3dx
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The integral of xn is n+1xn+1 when n=−1:
∫x3dx=4x4
So, the result is: 5670x4
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The integral of a constant times a function is the constant times the integral of the function:
∫(−20412x2)dx=−20412∫x2dx
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The integral of xn is n+1xn+1 when n=−1:
∫x2dx=3x3
So, the result is: −6804x3
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The integral of a constant times a function is the constant times the integral of the function:
∫10206xdx=10206∫xdx
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The integral of xn is n+1xn+1 when n=−1:
∫xdx=2x2
So, the result is: 5103x2
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The integral of a constant is the constant times the variable of integration:
∫(−2187)dx=−2187x
The result is: 16x8−192x7+1008x6−3024x5+5670x4−6804x3+5103x2−2187x
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Now simplify:
16(2x−3)8
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Add the constant of integration:
16(2x−3)8+constant
The answer is:
16(2x−3)8+constant
The answer (Indefinite)
[src]
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| 8
| 7 (2*x - 3)
| (2*x - 3) dx = C + ----------
| 16
/
16(2x−3)8
The graph
Use the examples entering the upper and lower limits of integration.