3.5.5—Inverse tangent

Use ∫du/(u²+a²)=(1/a)tan⁻¹(u/a)+C after rewriting the denominator as a square plus a positive constant.

Complete the square, factor the constant scale correctly, and use a substitution if the numerator contains the derivative of the shifted variable.

∫dx/[(x−2)²+9]=⅓tan⁻¹((x−2)/3)+C.

The coefficient is 1/a, not a; omitting it gives a derivative three times too large in the example.