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1. Homework Statement

1. Homework Statement

∫cosh

^{2}(x)sinh(x)dx = ?

## Homework Equations

cosh(x)sinh(x) = (1/2)*sinh(2x)

## The Attempt at a Solution

The solution is simple by guessing and checking with the chain rule:

∫cosh

^{2}(x)sinh(x)dx = 1/3*cosh

^{3}(x)

But then I try to manipulate the expression in the integral I get a different result:

cosh

^{2}(x)sinh(x) = cosh(x)cosh(x)sinh(x) = cosh(x)*(1/2)*sinh(2x) = (e

^{x}+e

^{-x})/2*(1/2)*(e

^{2x}-e

^{-2x})/2 = (1/4)*((e

^{3x}-e

^{-3x})/2+(e

^{x}-e

^{-x})/2 = (1/4)*(sinh(3x)+sinh(x))

Integrating that gives (1/4)*∫(sinh(3x)+sinh(x))dx = (1/4)*(∫sinh(3x)dx+∫sinh(x)dx) = (1/4)*((cosh(3x)/3)+cosh(x))

I'm not sure how to directly compare the two results, but plugging in some number for x and solving each shows that they aren't equal. I'm quite sure the first solution is correct, so where did I go wrong in the second?