2.3.2—An expression for x y d

When an integral contains a combination such as sinh x and cosh x, set u to a useful hyperbolic expression or use t=e^x. The identities cosh²x−sinh²x=1 and dx=dt/t can reduce the problem to algebraic or logarithmic terms.

Choose a substitution that matches the differential factor; do not expand blindly. Back-substitute and differentiate the result to verify the antiderivative.

With t=e^x, sinh x=(t−t⁻¹)/2 and cosh x=(t+t⁻¹)/2, so an integral in sinh and cosh becomes a rational expression in t.

A substitution is not complete until dx and the limits, if definite, are transformed consistently.