Math is Fun Website: Calculus | Integration Rules | Derivative Rules

Integration Examples

Common Functions	Function	Integral
Constant	∫a dx	ax + C
Variable	∫x dx	x2/2 + C
Square	∫x2 dx	x3/3 + C
Reciprocal	∫(1/x) dx	ln|x| + C
Exponential	∫ex dx	ex + C
	∫ax dx	ax/ln(a) + C
	∫ln(x) dx	x( ln(x) -1) + C
Trigonometry (x in radians)	∫cos(x) dx	sin(x) + C
	∫sin(x) dx	-cos(x) + C
	∫sec2(x) dx	tan(x) + C
Rules	Function	Integral
Multiplication by constant	∫cf(x) dx	c∫f(x) dx
Power Rule (n≠-1)	∫xn dx	xn+1/(n+1) + C
Sum Rule	∫(f + g) dx	∫f dx + ∫g dx
Difference Rule	∫(f - g) dx	∫f dx - ∫g dx
Integration by Parts	See Integration by Parts
Substitution Rule	See Integration by Substitution

Derivation Examples

Common Functions	Function	Derivative
Constant	c	0
	x	1
Square	x2	2x
Square Root	√x	½x-½
Exponential	ex	ex
	ax	ax(ln a)
Logarithms	ln(x)	1/x
	loga(x)	1 / (x ln(a))
Trigonometry (x is in radians)	sin(x)	cos(x)
	cos(x)	-sin(x)
	tan(x)	sec2(x)
	sin-1(x)	1/√(1-x2)
	tan-1(x)	1/(1+x2)
Rules	Function	Derivative
Multiplication by constant	cf	cf’
Power Rule	xn	nxn-1
Sum Rule	f + g	f’ + g’
Difference Rule	f - g	f’ - g’
Product Rule	fg	f g’ + f’ g
Quotient Rule	f/g	(f’ g - g’ f )/g2
Reciprocal Rule	1/f	-f’/f2
Chain Rule (as "Composition of Functions")	f º g	(f’ º g) × g’
Chain Rule (in a different form)	f(g(x))	f’(g(x))g’(x)