What is the integral of 2e^xsinh(x) ?

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

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integral of 2e^xsinh(x)

\int 2 e^{x} sinh (x) d x

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

Solution steps

\int 2 e^{x} sinh (x) d x

Take the constant out:

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

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

Apply Integration By Parts

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

= 2 (sinh (x) e^{x} - \int cosh (x) e^{x} d x)

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

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

Simplify

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

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

Add a constant to the solution

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

g (y) = - \frac{y}{( y ^{2} + 1 ) ^{\frac{3}{2}}}

\int ln (x^{2} + 15 x + 44) d x

\int \frac{1}{( 484 + x ^{2} ) ^{\frac{3}{2}}} d x

\frac{3 ^{2}}{( s + 3 ) ^{2}} \cdot \frac{1}{s}

\int \frac{1 - w}{w} d w

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