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

The integral of (sinh(x)-cosh(x))e^{-x} is -e^{-x}sinh(x)+C

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integral of (sinh(x)-cosh(x))e^{-x}

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

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

Solution steps

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

Expand

(sinh (x) - cosh (x)) e^{- x} : e^{- x} sinh (x) - e^{- x} cosh (x)

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

\int e^{- x} sinh (x) d x = - e^{- x} sinh (x) - \frac{1}{4} e^{- 2 x} + \frac{1}{2} x

\int e^{- x} cosh (x) d x = \frac{1}{2} (- \frac{1}{2} e^{- 2 x} + x)

= - e^{- x} sinh (x) - \frac{1}{4} e^{- 2 x} + \frac{1}{2} x - \frac{1}{2} (- \frac{1}{2} e^{- 2 x} + x)

Simplify

- e^{- x} sinh (x) - \frac{1}{4} e^{- 2 x} + \frac{1}{2} x - \frac{1}{2} (- \frac{1}{2} e^{- 2 x} + x) : - e^{- x} sinh (x)

= - e^{- x} sinh (x)

Add a constant to the solution

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

x^{2} d x - y (1 + x^{3}) d y = 0

x \to 0 lim (\frac{x ^{5} + 3 x}{x})

\int z (a + b z^{3})^{2} d z

x \to 0 lim (\frac{1}{x cot ( x )})

y = x (4.2)

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