What is the integral of (sinh(4x))/(e^{3x)} ?

The integral of (sinh(4x))/(e^{3x)} is 1/2 (1/7 e^{-7x}+e^x)+C

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integral of (sinh(4x))/(e^{3x)}

\int \frac{sinh ( 4 x )}{e ^{3 x}} d x

\frac{1}{2} (\frac{1}{7} e^{- 7 x} + e^{x}) + C

Solution steps

\int \frac{sinh ( 4 x )}{e ^{3 x}} d x

Rewrite using trig identities

= \int \frac{- 1 + e ^{8 x}}{2 e ^{7 x}} 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 \frac{- 1 + e ^{8 x}}{e ^{7 x}} d x

Expand

\frac{- 1 + e ^{8 x}}{e ^{7 x}} : - \frac{1}{e ^{7 x}} + e^{x}

= \frac{1}{2} \cdot \int - \frac{1}{e ^{7 x}} + e^{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

= \frac{1}{2} (- \int \frac{1}{e ^{7 x}} d x + \int e^{x} d x)

\int \frac{1}{e ^{7 x}} d x = - \frac{1}{7} e^{- 7 x}

\int e^{x} d x = e^{x}

= \frac{1}{2} (- (- \frac{1}{7} e^{- 7 x}) + e^{x})

Simplify

= \frac{1}{2} (\frac{1}{7} e^{- 7 x} + e^{x})

Add a constant to the solution

= \frac{1}{2} (\frac{1}{7} e^{- 7 x} + e^{x}) + C

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What is the integral of (sinh(4x))/(e^{3x)} ?
The integral of (sinh(4x))/(e^{3x)} is 1/2 (1/7 e^{-7x}+e^x)+C