What is the integral of (x^3-a^3)^2 ?

The integral of (x^3-a^3)^2 is (x^7)/7-(a^3x^4)/2+a^6x+C

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integral of (x^3-a^3)^2

\int (x^{3} - a^{3})^{2} d x

\frac{x ^{7}}{7} - \frac{a ^{3} x ^{4}}{2} + a^{6} x + C

Solution steps

\int (x^{3} - a^{3})^{2} d x

Expand

(x^{3} - a^{3})^{2} : x^{6} - 2 a^{3} x^{3} + a^{6}

= \int x^{6} - 2 a^{3} x^{3} + a^{6} 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 x^{6} d x - \int 2 a^{3} x^{3} d x + \int a^{6} d x

\int x^{6} d x = \frac{x ^{7}}{7}

\int 2 a^{3} x^{3} d x = \frac{a ^{3} x ^{4}}{2}

\int a^{6} d x = a^{6} x

= \frac{x ^{7}}{7} - \frac{a ^{3} x ^{4}}{2} + a^{6} x

Add a constant to the solution

= \frac{x ^{7}}{7} - \frac{a ^{3} x ^{4}}{2} + a^{6} x + C

What is the integral of (x^3-a^3)^2 ?
The integral of (x^3-a^3)^2 is (x^7)/7-(a^3x^4)/2+a^6x+C