Where did that come from?

[jimbod]

Can any one explain to me how the 1/3 in the second line is deduced from the equation above it in line one

 

[DrPhil]

Can any one explain to me how the 1/3 in the second line is deduced from the equation above it in line one

The second line is the antiderivative of the integrand. You can check that by differentiating the second line. You know that the derivative of x^3 is 3x^2. But the integrand is just -x^2, which is (-1/3)*(3x^2).

The antiderivative of \(\displaystyle x^n\) is \(\displaystyle \dfrac{1}{n+1}x^{n+1}\).

 

[HallsofIvy]

The second line is the antiderivative of the integrand. You can check that by differentiating the second line. You know that the derivative of x^3 is 3x^2. But the integrand is just -x^2, which is (-1/3)*(3x^2).

The antiderivative of \(\displaystyle x^n\) is \(\displaystyle (1/n)x^{n+1}\).

This is clearly a typo but a very unfortunate one since it contradicts the very point DrPhil is trying to make!

The antiderivative of \(\displaystyle x^n\) is \(\displaystyle (1/(n+1))x^{n+1}\).

 

[jimbod]

Thanks To both of you

Thanks to both of you I understand now. You were both a big help

This is clearly a typo but a very unfortunate one since it contradicts the very point DrPhil is trying to make!

The antiderivative of \(\displaystyle x^n\) is \(\displaystyle (1/(n+1))x^{n+1}\).