Power Rule for Integration

Derivatives

First let's look at the Power Rule for derivatives, one of the most commonly used rules in Calculus:

The derivative of xⁿ is nx⁽ⁿ⁻¹⁾

For x² we use the Power Rule with n=2:

Answer: the derivative of x² is 2x

So for derivatives we have:

Derivative diagram: x cubed goes to 3 times x squared by multiplying by power then reducing power by 1
"multiply by power
then reduce power by 1"

And for integration we reverse the rule:

Integration diagram: 3 times x squared goes to x cubed by adding 1 to power then dividing
"add 1 to power
then divide by the new power"

The integral of xⁿ is 1(n+1)x⁽ⁿ⁺¹⁾ + C

But not for n = −1, as that leads to division by zero
And C is the constant of integration. We add it because the derivative of any constant is zero, so there could have been a constant in the original function

The integral of x²	=	1(2+1)x²⁺¹ + C
	=	13x³ + C

Answer: ∫ x² dx = x³ 3 + C

√x is also x^0.5

∫x^0.5 dx = x^1.51.5 + C

= 23x^1.5 + C

Integration diagram: 3 times x squared goes to x cubed by adding 1 to power then dividing
"add 1 to power
then divide by the new power"

Here's the Power Rule for Integration with some sample values. See the pattern?

f(x)	∫ f(x) dx
1 (or x⁰)	x + C
x	12x² + C
x²	13x³ + C
x³	14x⁴ + C
x⁴	15x⁵ + C
and so on...
For negative exponents (except −1):
x⁻²	1−1x⁻¹ + C = −x⁻¹ + C
x⁻³	1−2x⁻² + C
and so on...

6824, 6825, 6834, 6835, 6827, 15373, 6833, 6839, 6840, 6842, 6843, 15367, 15375, 15376