Integration (Further Pure 2)
Standard integrals
$$\int \frac{1}{a^2 + x^2}\,dx = \frac{1}{a}\tan^{-1}\frac{x}{a} + C$$
$$\int \frac{1}{\sqrt{a^2 - x^2}}\,dx = \sin^{-1}\frac{x}{a} + C$$
- A trigonometric or hyperbolic substitution handles related forms.
Practice
What is ∫ 1/(a² + x²) dx?
This is the standard inverse-tangent integral.
Practice
What is ∫ 1/√(4 − x²) dx?
With a = 2, ∫1/√(a²−x²) dx = sin⁻¹(x/a) = sin⁻¹(x/2) + C.
Reduction & applications
- A reduction formula links $I_n$ to $I_{n-1}$, so you work down step by step.
- Integration also gives the arc length of a curve and the surface area of revolution.
Practice
A reduction formula links an integral Iₙ to Iₙ₋₁ so you can work down step by step.
Reduction formulae express a higher-index integral in terms of a lower one.
You've got it
Key idea
- $\int\dfrac{1}{a^2+x^2}dx = \dfrac1a\tan^{-1}\dfrac{x}{a} + C$; $\int\dfrac{1}{\sqrt{a^2-x^2}}dx = \sin^{-1}\dfrac{x}{a} + C$
- a reduction formula links $I_n$ to $I_{n-1}$
- integration also gives arc length and surface area of revolution