What is the integral of ((x^2-a^2)^4)/(x^2) ?

The integral of ((x^2-a^2)^4)/(x^2) is (x^7)/7-(4a^2x^5)/5+2a^4x^3-4a^6x-(a^8)/x+C

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integral of ((x^2-a^2)^4)/(x^2)

\int \frac{( x ^{2} - a ^{2} ) ^{4}}{x ^{2}} d x

\frac{x ^{7}}{7} - \frac{4 a ^{2} x ^{5}}{5} + 2 a^{4} x^{3} - 4 a^{6} x - \frac{a ^{8}}{x} + C

Solution steps

\int \frac{( x ^{2} - a ^{2} ) ^{4}}{x ^{2}} d x

Expand

\frac{( x ^{2} - a ^{2} ) ^{4}}{x ^{2}} : x^{6} - 4 a^{2} x^{4} + 6 a^{4} x^{2} - 4 a^{6} + \frac{a ^{8}}{x ^{2}}

= \int x^{6} - 4 a^{2} x^{4} + 6 a^{4} x^{2} - 4 a^{6} + \frac{a ^{8}}{x ^{2}} 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 4 a^{2} x^{4} d x + \int 6 a^{4} x^{2} d x - \int 4 a^{6} d x + \int \frac{a ^{8}}{x ^{2}} d x

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

\int 4 a^{2} x^{4} d x = \frac{4 a ^{2} x ^{5}}{5}

\int 6 a^{4} x^{2} d x = 2 a^{4} x^{3}

\int 4 a^{6} d x = 4 a^{6} x

\int \frac{a ^{8}}{x ^{2}} d x = - \frac{a ^{8}}{x}

= \frac{x ^{7}}{7} - \frac{4 a ^{2} x ^{5}}{5} + 2 a^{4} x^{3} - 4 a^{6} x - \frac{a ^{8}}{x}

Add a constant to the solution

= \frac{x ^{7}}{7} - \frac{4 a ^{2} x ^{5}}{5} + 2 a^{4} x^{3} - 4 a^{6} x - \frac{a ^{8}}{x} + C

What is the integral of ((x^2-a^2)^4)/(x^2) ?
The integral of ((x^2-a^2)^4)/(x^2) is (x^7)/7-(4a^2x^5)/5+2a^4x^3-4a^6x-(a^8)/x+C