# Question Video: Finding the Integration of a Rational Function by Using Integration by Substitution Mathematics • Higher Education

Determine β«((6π₯ + 8)/(3π₯Β² + 8π₯ + 3)) dπ₯.

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### Video Transcript

Determine the integral of six π₯ plus eight over three π₯ squared plus eight π₯ plus three with respect to π₯.

We will solve this problem using substitution. However, this will also lead us to a general rule that we can use for similar integration problems. Our first step is to let the denominator three π₯ squared plus eight π₯ plus three equal π’. Differentiating this would give us dπ’ by dπ₯. The differential of three π₯ squared is six π₯. And the differential of eight π₯ is eight.

Differentiating the constant three gives us zero. Therefore, dπ’ dπ₯ is equal to six π₯ plus eight. Rearranging this gives us that dπ’ is equal to six π₯ plus eight dπ₯. We can now replace the denominator of our initial expression with π’. And we can replace the numerator with dπ’. We are therefore left with the integral of one over π’ dπ’. This is the same as one of our standard integrals that we should know. The integral of one over π₯ with respect to π₯ is equal to ln π₯ plus π. This means that the integral of one over π’ with respect to π’ will be ln π’ plus π.

Our final step is to substitute our value for π’ back into the answer. The integral of six π₯ plus eight over three π₯ squared plus eight π₯ plus three is equal to ln of three π₯ squared plus eight π₯ plus three plus π. You might have noticed at the start that the top of our fraction was the differential of the bottom. Three π₯ squared plus eight π₯ plus three differentiated gives us six π₯ plus eight. If we are integrating π dash π₯ over π of π₯, our answer will always be the ln of π of π₯.

Another example that follows this method would be the integral of eight π₯ minus three over four π₯ squared minus three π₯ plus seven. Four π₯ squared minus three π₯ plus seven differentiates to eight π₯ minus three. This means that our answer would be ππ of four π₯ squared minus three π₯ plus seven plus π. To integrate a quotient when the numerator is the derivative of the denominator, we can use this rule.