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

The integral of ((x^2-a)^3)/2 is 1/2 ((x^7)/7-(3ax^5)/5+a^2x^3-a^3x)+C

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

integral

\frac{( x ^{2} - a ) ^{3}}{2}

\frac{1}{2} (\frac{x ^{7}}{7} - \frac{3 a x ^{5}}{5} + a^{2} x^{3} - a^{3} x) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int (x^{2} - a)^{3} d x

Expand

(x^{2} - a)^{3} : x^{6} - 3 a x^{4} + 3 a^{2} x^{2} - a^{3}

= \frac{1}{2} \cdot \int x^{6} - 3 a x^{4} + 3 a^{2} x^{2} - 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

= \frac{1}{2} (\int x^{6} d x - \int 3 a x^{4} d x + \int 3 a^{2} x^{2} d x - \int a^{3} d x)

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

\int 3 a x^{4} d x = \frac{3 a x ^{5}}{5}

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

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

= \frac{1}{2} (\frac{x ^{7}}{7} - \frac{3 a x ^{5}}{5} + a^{2} x^{3} - a^{3} x)

Add a constant to the solution

= \frac{1}{2} (\frac{x ^{7}}{7} - \frac{3 a x ^{5}}{5} + a^{2} x^{3} - a^{3} x) + C

What is the integral of ((x^2-a)^3)/2 ?
The integral of ((x^2-a)^3)/2 is 1/2 ((x^7)/7-(3ax^5)/5+a^2x^3-a^3x)+C