Integrate the following functions with respect to $x$:
- $5x^4+3x+10$
- $x^3-2x^2+9x$
For context I have recently completed my university degree and decided to go over some first year mathematics only to find i'm really struggling to remember integration, these are two questions that I have written down but I am struggling to remember the method behind it. I post this question hoping someone would be able to do some form of step-by-step calculation.
Do you know the power rule? $$ \frac{d}{dx}(ax^n)=anx^{n-1} $$ I remember this rule using the mantra 'multiply by the power, subtract one from the power'. Since integration is differentiation in reverse, we have to 'add one to the power, divide by the power', so $$ \int ax^n \, dx= \frac{ax^{n+1}}{n+1} + C\, . $$ Use the power rule, along with the sum rule, $$ \int f(x)+g(x) \, dx = \int f(x) \, dx + \int g(x) \, dx \, , $$ to solve your problem. You can write constants such as $10$ as $10x^{0}$.