Hector Petersen

2022-06-20

Help with Antiderivative (couple of questions)
I am not very good at this and still trying to understand how it works, but I really need to find a antiderivative of ${x}^{6}$. Would be very glad if someone could help me with that.

aletantas1x

Step 1
$\int {x}^{6}dx=\frac{{x}^{7}}{7}+C$
because $\frac{d}{dx}\left[\frac{{x}^{7}}{7}+C\right]=\frac{7{x}^{6}}{7}+0={x}^{6}$
Step 2
More generally, for all $n\ne -1$, $\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C$ because $\frac{d}{dx}\left[\frac{{x}^{n+1}}{n+1}+C\right]=\frac{\left(n+1\right){x}^{n}}{n+1}+0={x}^{n}$

rigliztetbf

Explanation:
The antiderivative of $\int$ simply using reverse power rule $\frac{{x}^{7}}{7}+C$
The rule is $\int$ ${x}^{n}$ $dx$ $=\frac{{x}^{n+1}}{n+1}$ $+C$.

Do you have a similar question?