What is the integral of sinh(3x+2) ?

The integral of sinh(3x+2) is 1/2 (1/3 e^{-3x-2}+1/3 e^{3x+2})+C

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integral of sinh(3x+2)

\int sinh (3 x + 2) d x

\frac{1}{2} (\frac{1}{3} e^{- 3 x - 2} + \frac{1}{3} e^{3 x + 2}) + C

Solution steps

\int sinh (3 x + 2) d x

Rewrite using trig identities

= \int \frac{1}{2} (- e^{- 3 x - 2} + e^{3 x + 2}) d x

Take the constant out:

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

= \frac{1}{2} \cdot \int - e^{- 3 x - 2} + e^{3 x + 2} 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

= \frac{1}{2} (- \int e^{- 3 x - 2} d x + \int e^{3 x + 2} d x)

\int e^{- 3 x - 2} d x = - \frac{1}{3} e^{- 3 x - 2}

\int e^{3 x + 2} d x = \frac{1}{3} e^{3 x + 2}

= \frac{1}{2} (- (- \frac{1}{3} e^{- 3 x - 2}) + \frac{1}{3} e^{3 x + 2})

Simplify

= \frac{1}{2} (\frac{1}{3} e^{- 3 x - 2} + \frac{1}{3} e^{3 x + 2})

Add a constant to the solution

= \frac{1}{2} (\frac{1}{3} e^{- 3 x - 2} + \frac{1}{3} e^{3 x + 2}) + C

What is the integral of sinh(3x+2) ?
The integral of sinh(3x+2) is 1/2 (1/3 e^{-3x-2}+1/3 e^{3x+2})+C