Find the integral(-x^8+4)^6 x^7

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Evaluate `int(-x^8+4)^6x^7dx` :

You can use the Fundamental Theorem of calculus directly:

Note that the derivative of `-x^8+4` is`-8x^7dx` , so we multiply the integrand by -8 and ` ``-1/8` to get:

`int -1/8((-x^8+4)^6-8x^7)dx` or `-1/8 int-8x^7(-x^8+4)^6dx`

This is of the form `int F(x)F'(x)dx` so by the FTC we get:

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`int (-x^8+4)^6x^7dx=-1/8int-8x^7(-x^8+4)^6dx`

`=-1/8[1/7(-x^8+4)^7+C_1]=-1/56(-x^8+4)^7+C` which is the solution.

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** You can check by differentiating:

`d/(dx)[-1/56(-x^8+4)^7+C=-1/56*7(-x^8+4)^6(-8x^7)`

`=-1/8(-8x^7)(-x^8+4)^6=(-x^8+4)^6x^7`

*** You could use a `u-` substitution:

Let `u=-x^8+4,(du)/(dx)=-8x^7`

Then `int(-x^8+4)^6x^7dx=-1/8intu^6du=-1/8(1/7u^7+C_1)`

`=-1/8(1/7(-x^8+4)^7+C_1=-1/56(-x^8+4)^7+C`

 

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