What is the integral of (x^3-1/(x^3))(x^3+1/(x^3)) ?

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

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integral of (x^3-1/(x^3))(x^3+1/(x^3))

\int (x^{3} - \frac{1}{x ^{3}}) (x^{3} + \frac{1}{x ^{3}}) d x

\frac{x ^{7}}{7} + \frac{1}{5 x ^{5}} + C

Solution steps

\int (x^{3} - \frac{1}{x ^{3}}) (x^{3} + \frac{1}{x ^{3}}) d x

Expand

(x^{3} - \frac{1}{x ^{3}}) (x^{3} + \frac{1}{x ^{3}}) : x^{6} - \frac{1}{x ^{6}}

= \int x^{6} - \frac{1}{x ^{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 \frac{1}{x ^{6}} d x

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

\int \frac{1}{x ^{6}} d x = - \frac{1}{5 x ^{5}}

= \frac{x ^{7}}{7} - (- \frac{1}{5 x ^{5}})

Simplify

= \frac{x ^{7}}{7} + \frac{1}{5 x ^{5}}

Add a constant to the solution

= \frac{x ^{7}}{7} + \frac{1}{5 x ^{5}} + C

\frac{d}{d u} (u^{2} v)

\int sin (\frac{x}{2}) cos (\frac{2 x}{3}) d x

81 m^{8} + 2 m^{4} + 1

arcsin (4 x + 1)

\frac{d}{d x} (\frac{x}{2 x ^{2} + 1})

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