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Methods of Integration
1.1 Basic formulas
xndx = xn+ n + 1
- C, n 6 = 1 ∫ exdx = ex^ + C;
x
dx = ln |x| + C ∫ cos xdx = sin x + C;
sin xdx = − cos x + C ∫ sec^2 xdx = tan x + C;
csc^2 xdx = − cot x + C ∫ sec x tan xdx = sec x + C;
csc x cot xdx = − csc x + C ∫ tan xdx = ln | sec x| + C;
cot xdx = ln | sin x| + C ∫ sec xdx = ln | sec x + tan x| + C;
csc xdx = ln | csc x − cot x| + C ∫ √ dx a^2 − x^2
= sin−^1 x a
+ C;
dx a^2 + x^2
a tan−^1 x a
+ C
dx x
x^2 − a^2
= cos−^1 a x
+ C
1.2 Trigonometric Integrals
Trigonometric identities:
- • cos^2 x + sin^2 x = 1 • sec^2 x = 1 + tan^2 x • csc^2 x = 1 + cot^2 x
- • cos^2 x = 1 + cos 2x 2 - sin^2 x = 1 − cos 2x 2 - cos x sin x = sin 2x 2
- • cos x cos y = 12 (cos(x + y) + cos(x − y))
- cos x sin y = 12 (sin(x + y) − sin(x − y))
- sin x sin y = 12 (cos(x − y) − cos(x + y))
Integral of the form
cosm^ x sinn^ xdx where m, n are non-negative integers,
Case 1. If m is odd, use cos xdx = d sin x. (Substitute u = sin x.)
Case 2. If n is odd, use sin xdx = −d cos x. (Substitute u = cos x.) Case 3. If both m, n are even, then use double angle formulas to reduce the power.
- cos^2 x = 1 + cos 2x 2 - sin^2 x = 1 − cos 2x 2 - cos x sin x = sin 2x 2
Integral of the form
secm^ x tann^ xdx where m, n are non-negative integers,
Case 1. If m is even, use sec^2 xdx = d tan x. (Substitute u = tan x.) Case 2. If n is odd, use sec x tan xdx = d sec x. (Substitute u = sec x.) Case 3. If both m is odd and n is even, use tan^2 x = sec^2 x − 1 to write everything in terms of sec x.
1.3 Integration By Parts
udv = uv −
vdu
1.4 Trigonometric Substitution
