How do you integrate #f(x)=5/(4x^3+4)# using the quotient rule?
The quotient rule applies to differentiation; not integration. The given function must be integrated using partial fraction expansion, variable substitution, and trigonometric substitution.
Note: I would have taken you, step-by-step, through the expansion but it made the explanation too long.
Break into two integrals:
Multiply the second integral by
Break the second integral into two integral so that the numerator of the first is
Simplify the third integral:
The first integral is the natural logarithm:
A variable substitution makes the second integral become the natural logarithm, too:
A trigonometric substitution makes the second integral become the inverse tangent: