Special forms and partial fractions

Six standard forms, all with a square (x² and a²) in the bottom:

• ∫dx/(x² + a²) = (1/a) tan⁻¹(x/a)
• ∫dx/(x² – a²) = (1/2a) log|(x – a)/(x + a)|
• ∫dx/(a² – x²) = (1/2a) log|(a + x)/(a – x)|
• ∫dx/√(a² – x²) = sin⁻¹(x/a)
• ∫dx/√(x² + a²) = log|x + √(x² + a²)|
• ∫dx/√(x² – a²) = log|x + √(x² – a²)|

Add + C to each.

Example: ∫dx/(x² + 9) = (1/3) tan⁻¹(x/3) + C.

If the bottom is a quadratic ax² + bx + c, complete the square to reach one of these. x² + 4x + 13 = (x + 2)² + 9, so ∫dx/(x² + 4x + 13) = (1/3) tan⁻¹((x + 2)/3) + C.

If the top is linear (px + q), write it as A × (derivative of the bottom) + B. The A part gives a log (top = derivative of bottom), and the B part is a standard form.

Partial fractions split a fraction with a factorised bottom into simple fractions you can integrate one by one:

Find A, B by putting convenient values of x. Example: x/((x – 1)(x – 2)) = A/(x – 1) + B/(x – 2). Then x = A(x – 2) + B(x – 1). x = 1 gives A = –1; x = 2 gives B = 2. So the integral is –log|x – 1| + 2 log|x – 2| + C.

This works only for a proper fraction (top degree less than bottom degree). If not, divide first.