Reciprocal-linear and logarithmic-derivative integrals — 9709 P3
Recognise logarithmic primitives and their domain. Identify a numerator proportional to the denominator derivative.
Notes and key methods
For \(\displaystyle b\ne 0, \int \frac{A}{bx+ c}dx= (\frac{A}{b})\ln |bx+ c|+ C\) on an interval that excludes the pole. Absolute values allow negative arguments of the inner linear function while keeping the real logarithm valid.
On an interval where bx+c>0, write ln(bx+c). A difference of endpoint logarithms can be combined as a logarithm of a positive ratio.
When \(\displaystyle g(x)\ne 0, \int \frac{g'(x)}{g(x)}dx= \ln |g(x)|+ C\). Compare the whole numerator with the derivative of the whole denominator before choosing a coefficient.
A positive quadratic denominator permits ln g(x) without absolute-value notation. Completing the square can establish positivity for all real x.
Check the conditions
Exclude the pole from the interval.
Use a logarithm rather than the ordinary power rule at exponent −1.
Compare the entire numerator and denominator derivative.
State an interval excluding any denominator zeros.
Three worked examples
Example 1 · Reciprocal-linear integrals
The integrand is \(\displaystyle f(x)=\frac{2}{2 x + 1}\), on \(\displaystyle x>- \frac{1}{2}\).
Find the general indefinite integral of the given function.
Evaluate the definite integral from \(0\) to \(1\).
Find the primitive G satisfying \(\displaystyle G(0)=0\).
Find the x-coordinate of the pole.
Show the reasoning
(a)
Differentiate \(\log{\left(2 x + 1 \right)}\) to recover the integrand. Include an arbitrary constant on each interval of the domain.
Answer: \(\log{\left(2 x + 1 \right)} + C\)
(b)
Use upper primitive value minus lower primitive value: \(\log{\left(3 \right)}\) − (\(0\)) = \(\log{\left(3 \right)}\).
Answer: \(\log{\left(3 \right)}\)
(c)
Subtract the primitive value at the initial abscissa, so the stated initial condition is satisfied.
Answer: \(G(x)=\log{\left(2 x + 1 \right)}\)
(d)
Set \(\displaystyle bx+c=0\).
\(\displaystyle - \frac{1}{2}\)
Answer: \(\displaystyle - \frac{1}{2}\).
Example 2 · The logarithmic-derivative pattern
The integrand is \(\displaystyle f(x)=\frac{6 x + 6}{x^{2} + 2 x + 5}\), on real x.
Find the general indefinite integral of the given function.
Evaluate the definite integral from \(0\) to \(1\).
Find the primitive G satisfying \(\displaystyle G(0)=0\).
Find the minimum value of the denominator.
Show the reasoning
(a)
Differentiate \(3 \log{\left(x^{2} + 2 x + 5 \right)}\) to recover the integrand. Include an arbitrary constant on each interval of the domain.