Search topics

Search

Anti-differentiate reciprocal and linear-reciprocal functions

Start a set of 8
Step 10 of 11

The rational integrand is shown below

\[q(z)=-4\,z + \frac{-6}{\left(-2\,z - 2\right)}\]
\[\int \left[-4\,z + \frac{-6}{\left(-2\,z - 2\right)}\right]\,dz\]

The stated branch of the linear denominator is shown below

\[-2\,z - 2 > 0\]
(a)1 mark

Find the full anti-derivative of the polynomial and reciprocal terms

The arbitrary additive constant is included in the family

(b)4 marks

Enter the coefficients of the polynomial primitive and the values that define the logarithmic term

Year 12, VCE Units 3 & 4, Chapter 5, 4 difficulty levels, No calculatorBuild a worksheet

Anti-differentiate reciprocal functions

The reciprocal of a linear expression has a logarithmic anti-derivative. Divide the numerator coefficient by the coefficient inside the denominator, use an absolute value on a general branch and include C for an indefinite integral

\[\int\frac{k}{ax+b}\,dx=\frac{k}{a}\log_e\left|ax+b\right|+C\]
  1. Divide by the inner coefficient Divide the numerator coefficient by the coefficient of x in the denominator \(\displaystyle \int\frac{6}{2x+1}\,dx=3\log_e\left|2x+1\right|+C\)
  2. Use an absolute value on a full branch The absolute value keeps the logarithm argument positive on either non-zero branch \(\displaystyle \int\frac{1}{x}\,dx=\log_e\left|x\right|+C\)
  3. Keep one arbitrary constant An indefinite integral represents a family of anti-derivatives, so include one arbitrary constant \(\displaystyle \int\left(f(x)+g(x)\right)\,dx=F(x)+G(x)+C\)
  4. Combine terminal logarithms For a definite reciprocal integral, combine the endpoint logarithms into one positive ratio \(\displaystyle \int_1^4\frac{2}{x+1}\,dx=2\log_e\left(\frac{5}{2}\right)\)

Worked example

Find the complete family of anti-derivatives of the reciprocal function

\[\int\frac{9}{3x-2}\,dx\]
  1. \(\displaystyle k=9,\quad a=3\)Identify the numerator coefficient and the coefficient of x in the denominator
  2. \(\displaystyle \frac{k}{a}=\frac{9}{3}=3\)Divide the numerator coefficient by the inner coefficient
  3. \(\displaystyle \int\frac{9}{3x-2}\,dx=3\log_e\left|3x-2\right|+C\)Use the reciprocal logarithm rule and include the arbitrary constant
  4. \(\displaystyle \frac{d}{dx}\left(3\log_e\left|3x-2\right|+C\right)=\frac{9}{3x-2}\)Differentiation confirms that the logarithmic expression has the required reciprocal derivative

Common mistakes

Missing the reciprocal inner coefficient
✕\(\displaystyle \int\frac{6}{2x+1}\,dx=6\log_e\left|2x+1\right|+C\)✓\(\displaystyle \int\frac{6}{2x+1}\,dx=3\log_e\left|2x+1\right|+C\)
The numerator coefficient must be divided by the inner coefficient 2 before it becomes the logarithmic coefficient
Omitting the absolute value
✕\(\displaystyle \int\frac{1}{x}\,dx=\log_e(x)+C\)✓\(\displaystyle \int\frac{1}{x}\,dx=\log_e\left|x\right|+C\)
The absolute value keeps the logarithm argument positive on either non-zero branch
Omitting the additive constant
✕\(\displaystyle \int\frac{4}{x-1}\,dx=4\log_e\left|x-1\right|\)✓\(\displaystyle \int\frac{4}{x-1}\,dx=4\log_e\left|x-1\right|+C\)
An indefinite integral represents a family, so one arbitrary constant must be included

Difficulty levels

introFind an indefinite logarithmic anti-derivative for a reciprocal with a constant numerator
standardFind a reciprocal anti-derivative with a linear denominator or evaluate a definite integral
challengePrepare a rational expression or use a boundary condition before applying the reciprocal rule
examChoose the correct reciprocal logarithm formula or apply it in a short multi-step question

Questions about this topic

Why is the coefficient of x divided out?+

Differentiating the linear expression gives its coefficient, so divide the numerator coefficient by that inner coefficient to recover the original numerator

When is the absolute value needed?+

Use the logarithm of the absolute value of the linear expression for a general non-zero branch. Where that expression is positive throughout the interval, the absolute value may be omitted

Why is there no C in a definite integral?+

The constant has the same value at both endpoints and cancels when the anti-derivative is evaluated in a definite integral

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Specialist Mathematics, chapter 5: Integral calculus

In this chapter

Integrate inverse circular kernels and change variablesAnti-differentiate reciprocal and linear-reciprocal functionsIntegration by recognition with logarithmic componentsAnti-differentiate by recognition from a supplied derivativeIntegrate using partial fractions, identities and partsDecompose rational expressions into partial fractionsCompound, double-angle and reciprocal identitiesAnti-differentiate exponential and circular functionsFind exact areas and assess numerical integration resultsFind the area between two curvesApproximate area by an endpoint or trapezium ruleFind areas involving reciprocal curvesAudit numerical integration loops and error enclosuresSet up volumes, parametric lengths and surfaces of revolution