02 The table of standard integrals (including trigonometric)
1cos x
Integral
- sin x + C
Comes from
- d/dx(sin x) = cos x
2sin x
Integral
- −cos x + C
Comes from
- d/dx(−cos x) = sin x
3sec² x
Integral
- tan x + C
Comes from
- d/dx(tan x) = sec² x
4cosec² x
Integral
- −cot x + C
Comes from
- d/dx(−cot x) = cosec² x
5sec x tan x
Integral
- sec x + C
Comes from
- d/dx(sec x) = sec x tan x
6cosec x cot x
Integral
- −cosec x + C
Comes from
- d/dx(−cosec x) = cosec x cot x
7tan x
Integral
- ln|sec x| + C
Comes from
- write tan x = sin x/cos x, u = cos x
8cot x
Integral
- ln|cosec x| + C
Comes from
- write cot x = cos x/sin x, u = sin x
9ex
Integral
- ex + C
Comes from
- self-derivative
101/x
Integral
- ln|x| + C
Comes from
- d/dx ln|x| = 1/x
Linear arguments
- Whenever the angle is (ax + b), divide by a: ∫ cos(ax + b) dx = sin(ax + b)/a + C, ∫ sin(ax + b) dx = −cos(ax + b)/a + C.
- ∫ sec²(2x) dx = tan(2x)/2 + C and ∫ e3x dx = e3x/3 + C — the same rule.
- Squares of trig functions are reduced with the double-angle identities first: sin²x = (1 − cos 2x)/2, cos²x = (1 + cos 2x)/2.
- Example: ∫ sin²x dx = ∫ (1 − cos 2x)/2 dx = x/2 − sin 2x/4 + C.