Integral of (2x+7)^8 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+7.
Then let du=2dx and substitute 2du:
∫2u8du
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The integral of a constant times a function is the constant times the integral of the function:
∫u8du=2∫u8du
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The integral of un is n+1un+1 when n=−1:
∫u8du=9u9
So, the result is: 18u9
Now substitute u back in:
18(2x+7)9
Method #2
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Rewrite the integrand:
(2x+7)8=256x8+7168x7+87808x6+614656x5+2689120x4+7529536x3+13176688x2+13176688x+5764801
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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:
∫256x8dx=256∫x8dx
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The integral of xn is n+1xn+1 when n=−1:
∫x8dx=9x9
So, the result is: 9256x9
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The integral of a constant times a function is the constant times the integral of the function:
∫7168x7dx=7168∫x7dx
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The integral of xn is n+1xn+1 when n=−1:
∫x7dx=8x8
So, the result is: 896x8
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The integral of a constant times a function is the constant times the integral of the function:
∫87808x6dx=87808∫x6dx
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The integral of xn is n+1xn+1 when n=−1:
∫x6dx=7x7
So, the result is: 12544x7
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The integral of a constant times a function is the constant times the integral of the function:
∫614656x5dx=614656∫x5dx
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The integral of xn is n+1xn+1 when n=−1:
∫x5dx=6x6
So, the result is: 3307328x6
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The integral of a constant times a function is the constant times the integral of the function:
∫2689120x4dx=2689120∫x4dx
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The integral of xn is n+1xn+1 when n=−1:
∫x4dx=5x5
So, the result is: 537824x5
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The integral of a constant times a function is the constant times the integral of the function:
∫7529536x3dx=7529536∫x3dx
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The integral of xn is n+1xn+1 when n=−1:
∫x3dx=4x4
So, the result is: 1882384x4
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The integral of a constant times a function is the constant times the integral of the function:
∫13176688x2dx=13176688∫x2dx
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The integral of xn is n+1xn+1 when n=−1:
∫x2dx=3x3
So, the result is: 313176688x3
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The integral of a constant times a function is the constant times the integral of the function:
∫13176688xdx=13176688∫xdx
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The integral of xn is n+1xn+1 when n=−1:
∫xdx=2x2
So, the result is: 6588344x2
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The integral of a constant is the constant times the variable of integration:
∫5764801dx=5764801x
The result is: 9256x9+896x8+12544x7+3307328x6+537824x5+1882384x4+313176688x3+6588344x2+5764801x
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Now simplify:
18(2x+7)9
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Add the constant of integration:
18(2x+7)9+constant
The answer is:
18(2x+7)9+constant
The answer (Indefinite)
[src]
/
| 9
| 8 (2*x + 7)
| (2*x + 7) dx = C + ----------
| 18
/
∫(2x+7)8dx=C+18(2x+7)9
The graph
9173533441
=
9173533441
Use the examples entering the upper and lower limits of integration.