What is the integral of sinh(5x+2) ?

The integral of sinh(5x+2) is 1/2 (1/5 e^{-5x-2}+1/5 e^{5x+2})+C

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integral of sinh(5x+2)

integral

sinh (5 x + 2)

\frac{1}{2} (\frac{1}{5} e^{- 5 x - 2} + \frac{1}{5} e^{5 x + 2}) + C

Solution steps

\int sinh (5 x + 2) d x

Rewrite using trig identities

= \int \frac{1}{2} (- e^{- 5 x - 2} + e^{5 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^{- 5 x - 2} + e^{5 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^{- 5 x - 2} d x + \int e^{5 x + 2} d x)

\int e^{- 5 x - 2} d x = - \frac{1}{5} e^{- 5 x - 2}

\int e^{5 x + 2} d x = \frac{1}{5} e^{5 x + 2}

= \frac{1}{2} (- (- \frac{1}{5} e^{- 5 x - 2}) + \frac{1}{5} e^{5 x + 2})

Simplify

= \frac{1}{2} (\frac{1}{5} e^{- 5 x - 2} + \frac{1}{5} e^{5 x + 2})

Add a constant to the solution

= \frac{1}{2} (\frac{1}{5} e^{- 5 x - 2} + \frac{1}{5} e^{5 x + 2}) + C

\frac{- 4}{( x ^{2} + 4 ) ^{2}}

\frac{3 x ^{2} + 4}{( x ^{3} + 4 x ) ^{4}}

\frac{3}{( u ^{2} - 2 u )}

\frac{3}{x + 1}

f (\frac{tan ( 1 )}{x})^{2}

What is the integral of sinh(5x+2) ?
The integral of sinh(5x+2) is 1/2 (1/5 e^{-5x-2}+1/5 e^{5x+2})+C