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

The integral of (e^{x/2}-e^{-x/2})^2 is e^xx-2x+e^{-x}x-2xcosh(x)+2sinh(x)+C

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integral of (e^{x/2}-e^{-x/2})^2

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

e^{x} x - 2 x + e^{- x} x - 2 x cosh (x) + 2 sinh (x) + C

Solution steps

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

Apply Integration By Parts

= (e^{\frac{x}{2}} - e^{- \frac{x}{2}})^{2} x - \int (e^{x} - e^{- x}) x d x

\int (e^{x} - e^{- x}) x d x = 2 x cosh (x) - 2 sinh (x)

= (e^{\frac{x}{2}} - e^{- \frac{x}{2}})^{2} x - (2 x cosh (x) - 2 sinh (x))

Simplify

(e^{\frac{x}{2}} - e^{- \frac{x}{2}})^{2} x - (2 x cosh (x) - 2 sinh (x)) : e^{x} x - 2 x + e^{- x} x - 2 x cosh (x) + 2 sinh (x)

= e^{x} x - 2 x + e^{- x} x - 2 x cosh (x) + 2 sinh (x)

Add a constant to the solution

= e^{x} x - 2 x + e^{- x} x - 2 x cosh (x) + 2 sinh (x) + C

d er i v a t i v e (x - 3) 4 - x

\frac{d}{d x} ((1 - x^{3})^{\frac{1}{2}})

t a y l or ln (x), 5

x \to - \infty lim (x^{n})

d er i v a t i v e F (x) = \frac{x ^{3} - 3 x ^{2} + 4 x - 5}{x ^{2}}

What is the integral of (e^{x/2}-e^{-x/2})^2 ?
The integral of (e^{x/2}-e^{-x/2})^2 is e^xx-2x+e^{-x}x-2xcosh(x)+2sinh(x)+C