Definite integral

Question

Please help me solve the two questions below

Define Q = ∫₃⁵ 2(x+1) ⁄ x²+6x+13

i) Rewrite Q into the form of ∫^b_a ms+x / s²+k²

Where a,b,c,k,m are constants

ii) Hence evaluate Q.

Answer 1

The given integral can be rewritten as ∫_{(from 3 to 5)}[(2x+6-4) / (x² + 6x + 13)]dx

= ∫_{(from 3 to 5)} [(2x+6) / (x²+6x+13)]dx - 4∫_{(from 3 to 5)}[1/(x²+6x+13)]dx

= ln(x²+6x+13)_{(from 3 to 5)} - 4∫_{(from 3 to 5)} [1 / ((x+3)² + 4)]dx (By completing the square)

= ln68 - ln40 - 4∫_{(from 6 to 8)} [ 1 / (u² + 2²) du] (Letting u = x+3]

= ln(17/10) - (4/2)Tan^-1(u/2)_{(from 6 to 8)}

= ln(17/10) - 2[Tan^-14 - Tan^-13]