Misapplying the power rule to fractional powers

The integration power rule says $\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+c$: raise the index by one, then divide by that new index. With a whole-number power this feels routine.

With a fractional power it wobbles in two places. First, students divide by the original index instead of the new one, or forget to add the one at all.

Second, the more damaging slip, they mishandle 'divide by a fraction'. Dividing by $\tfrac{3}{2}$ means multiplying by its reciprocal $\tfrac{2}{3}$, but under pressure many write $\times\tfrac{3}{2}$ instead, flipping the fraction the wrong way.

Roots and reciprocals make it worse, because $\sqrt{x}$ and $\tfrac{1}{\sqrt{x}}$ hide the index before it is even written as a power. The rule itself is short; the arithmetic of fractions is where the marks quietly leak away.

Integrate $\int\sqrt{x}\,dx$. First rewrite the root as a power: $\sqrt{x}=x^{1/2}$.

The new index is $\tfrac{1}{2}+1=\tfrac{3}{2}$, and the rule says divide by that new index.

Dividing by $\tfrac{3}{2}$ is the same as multiplying by $\tfrac{2}{3}$, so the coefficient should be $\tfrac{2}{3}$, not $\tfrac{3}{2}$.

Check by differentiating the answer, which should return the integrand. For the correct answer, $\frac{d}{dx}\left(\tfrac{2}{3}x^{3/2}\right)=\tfrac{2}{3}\cdot\tfrac{3}{2}x^{1/2}=x^{1/2}=\sqrt{x}$.

The wrong answer differentiates to $\tfrac{3}{2}\cdot\tfrac{3}{2}x^{1/2}=\tfrac{9}{4}\sqrt{x}$, which is not the integrand, so it fails the check.

1. 1

   Rewrite roots and reciprocals as powers

   Turn $\sqrt{x}$ into $x^{1/2}$ and $\tfrac{1}{\sqrt{x}}$ into $x^{-1/2}$ before touching the rule, so the index is visible.

2. 2

   Add one to the index

   Work out $n+1$ as a single fraction. For $n=\tfrac{1}{2}$, the new index is $\tfrac{3}{2}$; for $n=-\tfrac{1}{2}$, it is $\tfrac{1}{2}$.

3. 3

   Divide by the NEW index

   Never divide by the original index. Divide the power by $n+1$.

4. 4

   Turn 'divide by a fraction' into a multiplication

   Dividing by $\tfrac{3}{2}$ means $\times\tfrac{2}{3}$; dividing by $\tfrac{1}{2}$ means $\times 2$. Flip the fraction and multiply.

5. 5

   Add +c and differentiate to check

   For an indefinite integral include $+c$, then differentiate your answer, it should give back exactly what you started with.

Our teachers slow the fraction step right down, because that is where the marks go. We drill the same three moves, rewrite as a power, add one, divide by the new index, and say 'divide by three-halves means times two-thirds' out loud until the reciprocal stops flipping the wrong way.

In a one-to-one lesson we add the differentiation check so you can catch a wrong coefficient yourself before the paper does. Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English.

Message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.