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

The integral of (x^a-x^{-a})^2 is (x^{2a+1})/(2a+1)-2x+(x^{-2a+1})/(-2a+1)+C

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integral of (x^a-x^{-a})^2

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

\frac{x ^{2 a + 1}}{2 a + 1} - 2 x + \frac{x ^{- 2 a + 1}}{- 2 a + 1} + C

Solution steps

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

Expand

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

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

\int x^{2 a} d x = \frac{x ^{2 a + 1}}{2 a + 1}

\int 2 d x = 2 x

\int x^{- 2 a} d x = \frac{x ^{- 2 a + 1}}{- 2 a + 1}

= \frac{x ^{2 a + 1}}{2 a + 1} - 2 x + \frac{x ^{- 2 a + 1}}{- 2 a + 1}

Add a constant to the solution

= \frac{x ^{2 a + 1}}{2 a + 1} - 2 x + \frac{x ^{- 2 a + 1}}{- 2 a + 1} + C

What is the integral of (x^a-x^{-a})^2 ?
The integral of (x^a-x^{-a})^2 is (x^{2a+1})/(2a+1)-2x+(x^{-2a+1})/(-2a+1)+C