A student writes the integral of x to the 7th dx as x to the 8th over 3, plus 7. Find the antiderivative correctly and explain the error.

Step-by-step solution

1. 1

   State the power rule precisely. For every exponent $n\neq -1$,

   $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$

   Two separate things happen: the exponent goes up by one, and the term is divided by that new exponent $n+1$ — not by the old one, and not by anything else.

2. 2

   Apply it with $n=7$. The new exponent is $7+1=8$, so

   $\int x^7\,dx=\frac{x^8}{8}+C$

3. 3

   Locate the two separate errors in the proposal. In $\frac{x^8}{3}+7$ the exponent $8$ is right, but (i) the denominator should be $8$, and a $3$ appears from nowhere; and (ii) the constant is written as a specific number $7$, when an antiderivative must carry an arbitrary constant $C$ — every value of $C$ gives a valid antiderivative.

4. 4

   Differentiate the proposal to prove it is wrong. Differentiation is the built-in check for any integral:

   $\frac{d}{dx}\left(\frac{x^8}{3}+7\right)=\frac{8x^7}{3}$

   That is $\frac83$ times too large, so it cannot be an antiderivative of $x^7$.

5. 5

   Differentiate the correct answer.

   $\frac{d}{dx}\left(\frac{x^8}{8}+C\right)=\frac{8x^7}{8}=x^7$

   The $8$ in the denominator is exactly what cancels the $8$ the power rule for derivatives produces — which is why you divide by the new exponent. The answer is $\frac{x^8}{8}+C$.