# HOW TO INTEGRATE USING THE POWER RULE

The power rule of integration states that we should add one to the exponent of the variable and then divide by the value of the added exponent

That is,

$$\int x dx=\frac{x^{n+1}}{n+1}$$

STEPS TO INTEGRATE USING THE POWER RULE

1. Given an integral

2. Add one to the exponents of the variable of the integral

3. Divide by the sum of the variable of each exponent

4. Add C to the derived function to account for all possible solutions of the integral.

Example

Evaluate $\int 12x^5+10x^4+8x^3+6x^2+6 dx$

Solution:

Step 1: Given an integral

$\int 12x^5+10x^4+8x^3+6x^2+6 dx$

Step 2: Add one to the exponents of the variable of the integral.

$12x^{5+1}+10x^{4+1}+8x^{3+1}+6x^{2+1}+6x^{0+1}$

$12x^{6}+10x^{5}+8x^4+6x^3+6x^1$

Step 3: Divide by the sum of the exponent of each variable

$\frac{12x^{6}}{6}+\frac{10x^{5}}{5}+\frac{8x^4}{4}+\frac{6x^3}{3}+\frac{6x^1}{1}$

$2x^6+2x^5+2x^4+2x^3+6x$

Step 4: Add C to the derived function to account for all possible solutions of the integral

$2x^6+2x^5+2x^4+2x^3+6x+C$

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