What is the integral of (e^x-e^{-x})^3 ?

The integral of (e^x-e^{-x})^3 is 8(-cosh(x)+(cosh^3(x))/3)+C

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integral of (e^x-e^{-x})^3

\int (e^{x} - e^{- x})^{3} d x

8 (- cosh (x) + \frac{cosh ^{3} ( x )}{3}) + C

Solution steps

\int (e^{x} - e^{- x})^{3} d x

Rewrite using trig identities

= \int (2 sinh (x))^{3} d x

Simplify

(2 sinh (x))^{3} : 8 sinh^{3} (x)

= \int 8 sinh^{3} (x) d x

Take the constant out:

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

= 8 \cdot \int sinh^{3} (x) d x

Rewrite using trig identities

= 8 \cdot \int (- 1 + cosh^{2} (x)) sinh (x) d x

Apply u-substitution

: \int - 1 + u^{2} d u

= 8 \cdot \int - 1 + u^{2} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 8 (- \int 1 d u + \int u^{2} d u)

\int 1 d u = u

\int u^{2} d u = \frac{u ^{3}}{3}

= 8 (- u + \frac{u ^{3}}{3})

Substitute back

u = cosh (x)

= 8 (- cosh (x) + \frac{cosh ^{3} ( x )}{3})

Add a constant to the solution

= 8 (- cosh (x) + \frac{cosh ^{3} ( x )}{3}) + C

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What is the integral of (e^x-e^{-x})^3 ?
The integral of (e^x-e^{-x})^3 is 8(-cosh(x)+(cosh^3(x))/3)+C