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.

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

Beginner2022-06-21Added 22 answers

Step 1

$\int {x}^{6}dx=\frac{{x}^{7}}{7}+C$

because $\frac{d}{dx}[\frac{{x}^{7}}{7}+C]=\frac{7{x}^{6}}{7}+0={x}^{6}$

Step 2

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

$\int {x}^{6}dx=\frac{{x}^{7}}{7}+C$

because $\frac{d}{dx}[\frac{{x}^{7}}{7}+C]=\frac{7{x}^{6}}{7}+0={x}^{6}$

Step 2

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

rigliztetbf

Beginner2022-06-22Added 7 answers

Explanation:

The antiderivative of $\int $ simply using reverse power rule $\frac{{x}^{7}}{7}+C$

The rule is $\int $ ${x}^{n}$ $dx$ $=\frac{{x}^{n+1}}{n+1}$ $+C$.

The antiderivative of $\int $ simply using reverse power rule $\frac{{x}^{7}}{7}+C$

The rule is $\int $ ${x}^{n}$ $dx$ $=\frac{{x}^{n+1}}{n+1}$ $+C$.