Inverse-tangent integrals and mixed numerators
Pure 3 adds a direct antiderivative for a positive quadratic of the form . Its derivative check leads to inverse tangent.
Reverse the derivative of inverse tangent
One value of a controls both positions
denominator
x² + a²
↓
(1/a) tan⁻¹(x/a) + C
divide outside · divide the input
If a² = 25, then a = 5 — not 25.
Here . Read as the positive square root of the constant beside .
The formula follows from the inverse-tangent derivative. If , then . Therefore:
Worked example
Normalize the denominator before reading a
Now . The outside factors cancel:
Answer
Check by differentiating the answer; the result is .
(Pure Mathematics 2 & 3 Coursebook, Ch. 8.1–8.2)
Split a linear numerator into two jobs
Over , an -term can produce a logarithm while a constant term produces an inverse tangent.
Answer
Common mistake
Do not choose between logarithm and inverse tangent for the whole numerator. Split it: the two parts can require different methods.
Your turnIndependent check [5 marks]
Find .
Show worked answer
Answer