From Radicals to Fractional Powers
A radical can make an expression look different from a power, but the two notations describe the same kinds of quantities. Rewriting a radical as a fractional power lets us use the power rule for antiderivatives, which already works for powers of the variable.
For \(x>0\), a fractional power is defined by taking a root and, if needed, a whole-number power. For example, \(x^{1/2}=\sqrt{x}\), while \(x^{3/2}=(\sqrt{x})^3\). A negative exponent means reciprocal: \(x^{-1/2}=1/x^{1/2}=1/\sqrt{x}\).
The central rule is a direct extension of the power rule for antiderivatives: increase the exponent by one, then divide by the new exponent. The exponent must not be \(-1\). The special case with exponent \(-1\) has a logarithmic antiderivative and is treated separately.
Why the Rule Works
An antiderivative of a function \(f\) is a function \(F\) whose derivative is \(f\). To justify the rule, differentiate its proposed right-hand side. The derivative power rule applies for \(x>0\) and real exponent \(r+1\), and the constant \(1/(r+1)\) does not depend on \(x\).
The cancellation requires \(r+1\ne0\), which is exactly the condition \(r\ne-1\). Thus the proposed function differentiates to the original power, so it is an antiderivative. The \(+C\) accounts for the fact that adding any constant does not change the derivative.
This verification is a useful habit: after finding an antiderivative, differentiate the result and check that the original integrand returns. It catches errors in both the exponent and the coefficient.
Worked Examples
Worked Example: Integrate a Square Root
Find an antiderivative of \(\sqrt{x}\) for \(x>0\). First rewrite the radical as a power:
Here the exponent is \(r=1/2\), which is not \(-1\). Increase it by one and divide by the new exponent:
Check by differentiation. The coefficient \(2/3\) multiplies the derivative coefficient \(3/2\), and the exponents reduce from \(3/2\) to \(1/2\):
The derivative matches the integrand, so \(\frac{2}{3}x^{3/2}+C\) is the family of antiderivatives on \(x>0\).
Worked Example: Integrate a Reciprocal Square Root
Find an antiderivative of \(1/\sqrt{x}\) for \(x>0\). Rewrite the reciprocal radical with a negative fractional exponent:
Now apply the rule with \(r=-1/2\). The new exponent is \(-1/2+1=1/2\), so the divisor is \(1/2\):
Differentiate the answer to check both the coefficient and the negative exponent:
The result differentiates to the original reciprocal square root.
Worked Example: Integrate a Higher Fractional Power
Find an antiderivative of \(x\sqrt{x}\) for \(x>0\). Convert the radical, then combine the powers using \(x^a x^b=x^{a+b}\):
The exponent \(3/2\) is not \(-1\). Increasing it by one gives \(5/2\), and dividing by \(5/2\) is the same as multiplying by \(2/5\):
Differentiate to verify the answer:
The differentiated expression agrees with the original integrand.
A Reliable Method
For an integrand involving a radical or reciprocal radical, the main work is often rewriting it accurately. Once it is a power of \(x\), the antiderivative rule can be applied. The following sequence makes the steps explicit.
Express each root as a fractional power and each reciprocal as a negative power.
Combine powers of \(x\) when they are multiplied, using the laws of exponents.
If it is \(-1\), this power rule does not apply. Otherwise, add one to the exponent and divide by the result.
Include the arbitrary constant, then differentiate your proposed antiderivative to recover the integrand.
Common Mistake: Forgetting to Divide
A frequent error is to increase the exponent but leave out the coefficient created by dividing by the new exponent. For instance, a wrong calculation for \(\int x^{1/2}\,dx\) is:
Differentiate the proposed answer to test it:
The derivative is too large by a factor of \(3/2\). The correction is to divide by the new exponent \(3/2\), or equivalently multiply by \(2/3\):
Its derivative is \(\frac{2}{3}\cdot\frac{3}{2}x^{1/2}=x^{1/2}\), as required. A second useful check is that the exponent must increase by one; changing \(x^r\) to \(x^{r-1}\) is not the antiderivative power rule.
When the Rule Does Not Apply
The condition \(r\ne-1\) matters. If a rewritten integrand has exponent \(-1\), adding one would give zero in the denominator, so the displayed power rule is undefined. For example, \(1/x=x^{-1}\) is not integrated by substituting \(r=-1\) into this formula. Its antiderivative uses a different rule, which is considered separately.
Also keep track of the domain. A reciprocal radical such as \(1/\sqrt{x}\) requires \(x>0\): the square root is not real for negative \(x\), and its reciprocal is undefined at zero. For \(\sqrt{x}\), the expression itself is defined at zero, but the rule and derivative verification above are stated on \(x>0\). These domain conditions ensure the expressions used in the calculation are defined where the antiderivative is being considered.
Why Rewriting Matters
Recognizing equivalent forms is more than a change in notation. It reveals which rule applies. The expressions \(\sqrt{x}\), \(1/\sqrt{x}\), and \(x\sqrt{x}\) look different, but their power forms are \(x^{1/2}\), \(x^{-1/2}\), and \(x^{3/2}\). Each then fits the same method.
When an answer seems plausible, do not rely on its appearance. Differentiate it. A correct antiderivative must return the original function, including its sign, coefficient, and exponent. That check is especially valuable when negative fractional exponents or fractions such as \(2/5\) appear.
Check Your Understanding
For each question, rewrite radicals as powers where needed and use differentiation to check your result.
- Find an antiderivative of \(\sqrt[3]{x}\) on \(x>0\).
- Rewrite \(1/\sqrt[3]{x^2}\) as a power of \(x\), then find an antiderivative on \(x>0\).
- Why does the power rule for antiderivatives require \(r\ne-1\)?
- Find an antiderivative of \(x^2\sqrt{x}\) on \(x>0\).
- What goes wrong if you increase the exponent but do not divide by the new exponent?
