What is the integral of (e^{-2}x)sinh^2(x) ?

The integral of (e^{-2}x)sinh^2(x) is 1/(8e^2)(-2x^2+2xsinh(2x)-cosh(2x))+C

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integral of (e^{-2}x)sinh^2(x)

integral

(e^{- 2} x) sinh^{2} (x)

\frac{1}{8 e ^{2}} (- 2 x^{2} + 2 x sinh (2 x) - cosh (2 x)) + C

Solution steps

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

Take the constant out:

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

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

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{e ^{2}} \cdot \int x sinh^{2} (x) d x

Apply Integration By Parts

: \frac{1}{2} x (- x + \frac{1}{2} sinh (2 x)) - \int \frac{1}{2} (- x + \frac{1}{2} sinh (2 x)) d x

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

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

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

Simplify

\frac{1}{e ^{2}} (\frac{1}{2} x (- x + \frac{1}{2} sinh (2 x)) - \frac{1}{2} (- \frac{x ^{2}}{2} + \frac{1}{4} cosh (2 x))) : \frac{1}{8 e ^{2}} (- 2 x^{2} + 2 x sinh (2 x) - cosh (2 x))

= \frac{1}{8 e ^{2}} (- 2 x^{2} + 2 x sinh (2 x) - cosh (2 x))

Add a constant to the solution

= \frac{1}{8 e ^{2}} (- 2 x^{2} + 2 x sinh (2 x) - cosh (2 x)) + C

\frac{x}{( 2 x ^{2} + 1 )}

sin^{3} (2 x + 5)

3 x^{2} (x^{3} + 1)^{10}

x^{2} lo g_{10} (1 + x)

x^{2} e^{l n (x) + x}

What is the integral of (e^{-2}x)sinh^2(x) ?
The integral of (e^{-2}x)sinh^2(x) is 1/(8e^2)(-2x^2+2xsinh(2x)-cosh(2x))+C