Just as you have a power rule in differentiation, you have one in integration. Further, since integration is exactly the opposite of differentiation, the method you use while integrating simple powers of 'x' is exactly the opposite while differentiating them.
Recall that while differentiating simple powers of `x`, like `x^2`, you multiply with the power itself (which is 2 here) and then decrease it by 1. Thus `d/dx x^2 = 2x`. In integration, you increase the power by 1 and divide by the new number. So `\int x^2 = x^3/3 + C`.
The power rule of integration can be understood in two steps:
Note: The power rule can not be used to integrate`x^-1`. This is because when you increase -1 by 1, you get 0. Thus you get `\int x^-1 dx = x^0 / 0` in which, as you can see, there is division by zero. Division by zero does not give a well defined answer. This rule is used instead.
Recall that while differentiating simple powers of `x`, like `x^2`, you multiply with the power itself (which is 2 here) and then decrease it by 1. Thus `d/dx x^2 = 2x`. In integration, you increase the power by 1 and divide by the new number. So `\int x^2 = x^3/3 + C`.
The power rule of integration can be understood in two steps:
- Increase the power (exponent) by 1
- Divide by the new power (exponent)
`\int x^n dx= x^(n+1)/(n+1) + C`... where 'n' is any real number except -1 (that is, it can be any negative or positive number except -1 and even a fraction, an irrational number or a decimal)
Note: The power rule can not be used to integrate`x^-1`. This is because when you increase -1 by 1, you get 0. Thus you get `\int x^-1 dx = x^0 / 0` in which, as you can see, there is division by zero. Division by zero does not give a well defined answer. This rule is used instead.
Examples
1. `\int x^100 dx``= x^(100 + 1)/(100 + 1) + C `
`= (x^101)/101 + C`2. `\int x^(-10) dx`
`= x^(-10 + 1) / (-10 + 1) + C`
`= x^-9 / -9 + C`3. `\int 1/x^2 dx`
`= \int x^-2 dx`
`= x^(-2+1) / (-2 + 1) + C`
`= x^(-1) / (-1) + C`
` = -1/x + C`4. `\int x^(2/3) dx`
`= x^(2/3 + 1) / (2/3 + 1) + C`
`= x^(5/3) / (5/3) + C`
`= 3/5 * x^(5/3) + C`5. `\int sqrt(x) dx`
`= \int x^(1/2) dx`
`= x^(1/2 + 1) / (1/2 + 1) + C`
`= x^(3/2) / (3/2) + C`
`= 2/3*x^(3/2) + C`6. `\int 1/sqrt(x) dx`
`= \int x^(-1/2) dx`
`= x^(-1/2 + 1) / (-1/2 + 1) + C`
`= x^(1/2) / (1/2) + C`
`= 2*sqrt(x) + C`7. `\int x^log(2) dx`
`= \int x^(log(2) + 1) / (log(2) + 1) + C` ( log(2) is a number )8. `\int x^sqrt(3) dx`
`= \int x^(sqrt(3) + 1) / (sqrt(3) + 1) + C`Next Post: Integrating 1/x
