Integration 1

INTEGRATION SEBASTIAN VATTAMATTAM 1. Basic Formulae (1.1) Z dx = x + c Z xndx = xn+1 n + 1 (1.2) + c, n 6= 1 Z 1x(1.3) dx = log(|x|) + c (1.4) Z exdx = ex + c (1.5) Z sin(x)dx = −cos(x) + c (1.6) Z cos(x)dx = sin(x) + c (1.7) Z sec2(x)dx = tan(x) + c (1.8) Z csc2(x)dx = −cot(x) + c (1.9) Z sec(x) tan(x)dx = sec(x) + c (1.10) Z csc(x) cot(x)dx = −csc(x) + c Z 1 p1 − x2 (1.11) dx = sin−1(x) + c Z 1 1 + x2 (1.12) dx = tan−1(x) + c Z 1 xp1 − x2 (1.13) dx = sec−1(x) + c 2. Solved Problems Example 2.1. Integrate w.r.t x, y = 2x2 − 4x + 3 px 12 SEBASTIAN VATTAMATTAM Z ydx = Z [2x2 px − 4x px + 3 px] = 2 Z x3/2dx − 4 Z x1/2dx + 3 Z x−1/2dx = 2x5/2 5/2 − 4x3/2 3/2 + 3x1/2 1/2 + c = 45x5/2 − 83x3/2 + 6px + c Example 2.2. Evaluate Z 1 1 − sin xdx Z 1 1 − sin xdx = Z 1 + sin x (1 − sin x)(1 + sin x)dx = Z 1 + sin x (cos2 x) dx = Z [sec2 x + sec x tan x]dx = tan x + sec x + c Example 2.3. Try to evaluate Z p1 − sin 2x