Question

Find the integral. \(\int \cosh ^{2}(x-1) \sinh (x-1) d x\)

Short answer

The integral of \( \cosh^2(x-1) \sinh(x-1) \) is \( \cosh^2(x-1)^2 + c \)

Step-by-step solution

Rewrite the Integral

First, recognize that \(\sinh(x-1)\) is the derivative of \(\cosh(x-1)\). So we can rewrite the integral as \(\int u du\) where \(u = \cosh^2(x-1)\). Then \(\frac{du}{dx} = 2 \cosh(x-1) \sinh(x-1)\). Notice that we can now cancel out the \(\sinh(x-1)\) terms.

Change of Variables

To perform a change of variables, it might be easier to bring the constant factor of 2 outside of the integral. Now we have \(2 \int \cosh(x-1) du\). Referring to our substitution, this just turns into \(2u du\).

Compute the Integral

Evaluate the integral, leaving us with: \( u^2+c \). Substitute back for \( u \) giving us the final answer \(\cosh^2(x-1)^2 + c \).