What is the integral of (8x-a)^3 ?

The integral of (8x-a)^3 is 128x^4-64ax^3+12a^2x^2-a^3x+C

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integral of (8x-a)^3

\int (8 x - a)^{3} d x

128 x^{4} - 64 a x^{3} + 12 a^{2} x^{2} - a^{3} x + C

Solution steps

\int (8 x - a)^{3} d x

Expand

(8 x - a)^{3} : 512 x^{3} - 192 a x^{2} + 24 a^{2} x - a^{3}

= \int 512 x^{3} - 192 a x^{2} + 24 a^{2} x - a^{3} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 512 x^{3} d x - \int 192 a x^{2} d x + \int 24 a^{2} x d x - \int a^{3} d x

\int 512 x^{3} d x = 128 x^{4}

\int 192 a x^{2} d x = 64 a x^{3}

\int 24 a^{2} x d x = 12 a^{2} x^{2}

\int a^{3} d x = a^{3} x

= 128 x^{4} - 64 a x^{3} + 12 a^{2} x^{2} - a^{3} x

Add a constant to the solution

= 128 x^{4} - 64 a x^{3} + 12 a^{2} x^{2} - a^{3} x + C

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What is the integral of (8x-a)^3 ?
The integral of (8x-a)^3 is 128x^4-64ax^3+12a^2x^2-a^3x+C