Evaluate the indefinite integral `int x(8x+7)^8 dx`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The integral `int x*(8x+7)^8 dx` has to be determined.

Let `8x + 7 = y`

=> `x = (y - 7)/8`

`dx = (1/8)*dy`

`int x*(8x+7)^8 dx`

=> `(1/8)*int (y - 7)/8*y^8 dy`

=> `(1/64)*int y^9 - 7*y^8 dy`

=> `(1/64)*(y^10/10 - (7*y^9)/9)`

=> `y^10/640 - (7*y^9)/576`

substitute `y = 8x + 7`

=> `(8x+7)^10/640 - (7*(8x+7)^9)/576`

The integral `int x*(8x+7)^8 dx = (8x+7)^10/640 - (7*(8x+7)^9)/576 + C`

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