Math Integration rules Sum rule and difference rule

# Sum rule and difference rule

The Sum- and difference rule states that a sum or a difference is integrated termwise.

## Sum rule

$\int (f(x)+g(x)) \, \mathrm{d}x =$ $\int f(x) \, \mathrm{d}x + \int g(x) \, \mathrm{d}x$

### Example

$f(x)=x^4+x^3$

1. #### Split function into subfunctions

$g(x)=x^4$ and $h(x)=x^3$
2. #### Integrate subfunctions

$\int x^4=\frac15x^5$ and $\int x^3=\frac14x^4$
3. $\int (x^4+x^3)$ $=\int x^4+ \int x^3$ $=\frac15x^5+\frac14x^4\color{purple}{+C}$
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### Hint

If it is, like in this case, an indefinite integral, then the constant of integration $\color{purple}{C}$ must be set at the end.

## Difference rule

$\int (f(x)-g(x)) \, \mathrm{d}x =$ $\int f(x) \, \mathrm{d}x - \int g(x) \, \mathrm{d}x$

### Example

$f(x)=x^4-x^3$

1. #### Split function into subfunctions

$g(x)=x^4$ und $h(x)=x^3$
2. #### Integrate subfunctions

$\int x^4=\frac15x^5$ and $\int x^3=\frac14x^4$
3. $\int (x^4-x^3)$ $=\int x^4- \int x^3$ $=\frac15x^5-\frac14x^4\color{purple}{+C}$