What is the integral of sinh^2(x) dx?

What is the integral of sinh^2(x) dx?

• A. (1/2)(x - sinh(x)cosh(x)) + C
• B. (1/2)e^x + C
• C. sinh(x) + C
• D. (1/2)(sinh(2x)) + C

Answer: (A)

To find the integral of \( \sinh^2(x) \, dx \), it's useful to apply a hyperbolic identity to simplify the integration process. The identity states: \[ \sinh^2(x) = \frac{1}{2}(\cosh(2x) - 1) \] This allows us to rewrite the integral as: \[ \int \sinh^2(x) \, dx = \int \frac{1}{2} (\cosh(2x) - 1) \, dx \] Breaking this down further, we integrate each term separately: 1. The integral of \( \cosh(2x) \) is \( \frac{1}{2} \sinh(2x) \). 2. The integral of \( 1 \) is simply \( x \). Putting these results together, we have: \[ \int \sinh^2(x) \, dx = \frac{1}{2} \left( \frac{1}{2} \sinh(2x) - x \right) + C = \frac{1}{2} \sinh(2x) - \frac{1

To find the integral of ( \sinh^2(x) , dx ), it's useful to apply a hyperbolic identity to simplify the integration process. The identity states:

[

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

]

This allows us to rewrite the integral as:

[

\int \sinh^2(x) , dx = \int \frac{1}{2} (\cosh(2x) - 1) , dx

]

Breaking this down further, we integrate each term separately:

1. The integral of ( \cosh(2x) ) is ( \frac{1}{2} \sinh(2x) ).

2. The integral of ( 1 ) is simply ( x ).

Putting these results together, we have:

[

\int \sinh^2(x) , dx = \frac{1}{2} \left( \frac{1}{2} \sinh(2x) - x \right) + C = \frac{1}{2} \sinh(2x) - \frac{1