# 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:

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:

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To integrate ( f(x) = \frac{5}{4x^3 + 4} ) using the quotient rule, you first express the function as the quotient of two functions, ( u(x) ) and ( v(x) ). In this case, ( u(x) = 5 ) and ( v(x) = 4x^3 + 4 ). Then, you apply the quotient rule formula:

[ \int \frac{u(x)}{v(x)} , dx = \ln|v(x)| + C ]

where ( C ) is the constant of integration. Therefore,

[ \int \frac{5}{4x^3 + 4} , dx = \frac{5}{4} \int \frac{1}{x^3 + 1} , dx ]

To integrate ( \frac{1}{x^3 + 1} ), you can use the substitution method with ( u = x^3 + 1 ), then ( du = 3x^2 , dx ), and rewrite the integral as:

[ \frac{1}{3} \int \frac{1}{u} , du ]

Integrating ( \frac{1}{u} ) with respect to ( u ) gives ( \ln|u| ). Substituting back for ( u ) gives:

[ \frac{1}{3} \ln|x^3 + 1| + C ]

So, the integral of ( f(x) ) is:

[ \frac{5}{12} \ln|x^3 + 1| + C ]

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