C4 Integration

Part 1 – Integrating more complicated functions C4 Integration Part 1 – Integrating more complicated functions

Integrating Standard Functions

Ex 6A page 83

Reversing the chain rule – integration by recognition This works for functions of the form f(ax + b) For example; integrate w.r.t.x cos(4x - 1) Start by thinking what would y = sin (4x - 1) differentiate into? 4cos(4x - 1) So ∫cos(4x – 1) = ¼ sin(4x – 1) + C

Reversing the chain rule – integration by recognition Example 2 For example; integrate w.r.t.x (3x + 4)3 Start by thinking what would y = (3x + 4)4 differentiate into? 4 x 3(3x + 4)3 So ∫ (3x + 4)3 = 1/12 (3x + 4)4 + C

Reversing the chain rule – integration by recognition Example 3 For example; integrate w.r.t.x e2x+3 Start by thinking what would y = e2x+3 differentiate into? 2 e2x+3 So ∫ e2x+3 = ½ e2x+3 + C

Reversing the chain rule – integration by recognition Example 4 For example; integrate w.r.t.x sec4xtan4x Start by thinking what would y = tan 4x differentiate into? What about y = sec 4x sec 4x differentiates to 4sec4xtan4x So ∫ sec 4x tan 4x = ¼ sec 4x + C

Ex 6B page 86

Using trig identites in integration From C2? sin2x + cos2x = 1 Divide by cos2x tan2x + 1 = sec2x