Why Look Inside the Radical?
A square root can make an integral look more complicated than it is. When the expression under the root is itself a function, the key question is whether the numerator contains the derivative of that inner function. If it does, substitution can replace the whole expression under the root with a single variable.
For instance, in an integral involving \(\sqrt{3x^2+2}\), the inner expression is \(3x^2+2\), whose derivative is \(6x\). If the integral also contains \(6x\) as a factor, the substitution \(u=3x^2+2\) converts the radical into \(\sqrt{u}\) and the remaining factor into \(du\).
This tutorial develops that pattern for square roots, reciprocal square roots, and other powers of a square root. The method is an application of substitution as the reverse chain rule: the inner function and its derivative must appear together.
Rewrite the Radical as a Power
For a positive real number \(u\), the square root can be written as \(u^{1/2}\), and its reciprocal can be written as \(u^{-1/2}\). More generally, a reciprocal power of a square root becomes a negative power of the radicand. These rewrites make the power rule available after substitution.
The positivity condition matters when a reciprocal root is present: \(1/\sqrt{g(x)}\) is defined only when \(g(x)>0\). We will work on intervals where \(g\) is differentiable and positive, so all the displayed powers and derivatives are defined.
With \(u=g(x)\), the differential relation is \(du=g'(x)\,dx\). If the integrand has a constant multiple of \(g'(x)\), that constant must remain in the calculation. Once everything is expressed in \(u\), integrate using the power rule.
For example, \(\sqrt{g(x)}=[g(x)]^{1/2}\), while \(1/\sqrt{g(x)}=[g(x)]^{-1/2}\). A reciprocal cube of the square root is \([g(x)]^{-3/2}\). These exponents are not all the same, so it is important to rewrite the exact integrand rather than assume every radical integral has the same antiderivative pattern.
Why the Substitution Rule Works
The rule follows directly from the chain rule. Suppose \(g(x)>0\), and let \(p\ne-1\). The outer function \(u^{p+1}/(p+1)\) has derivative \(u^p\). Composing it with \(g(x)\) gives an antiderivative after multiplying by the derivative of the inner function.
The derivative of \(u^{p+1}/(p+1)\) is \(u^p\), since the factor \(p+1\) cancels.
Substituting \(g(x)\) for \(u\) contributes the inner derivative \(g'(x)\).
Multiplying by \(c\) gives the desired integrand \(c\,g'(x)[g(x)]^p\).
In equations, the verification is:
This proves the formula on any interval where the indicated powers are defined and the chain rule applies. The condition \(p\ne-1\) is needed because the displayed antiderivative divides by \(p+1\).
Worked Examples
Worked Example: Integrating a Square Root
Evaluate \(\int 6x\sqrt{3x^2+2}\,dx\).
The inner expression is \(3x^2+2\). Its derivative is \(6x\), which matches the factor outside the square root. Set \(u=3x^2+2\), so \(du=6x\,dx\). Also, \(\sqrt{3x^2+2}=u^{1/2}\). Therefore:
Check by differentiating the answer:
The derivative equals the original integrand, so the antiderivative is correct.
Worked Example: Integrating a Reciprocal Square Root
Evaluate \(\int \frac{4x}{\sqrt{2x^2+5}}\,dx\).
Let \(u=2x^2+5\). Then \(du=4x\,dx\), and the reciprocal square root becomes \(u^{-1/2}\). Thus:
To check, differentiate the result and apply the chain rule:
This matches the integrand. The radicand \(2x^2+5\) is positive for every real \(x\), so the reciprocal root is defined on all of \(\mathbb{R}\).
Worked Example: A Reciprocal Power of a Square Root
Evaluate \(\int \frac{6x}{(\sqrt{3x^2+1})^3}\,dx\).
Here the square root is raised to the third power, so the denominator is \((3x^2+1)^{3/2}\). Set \(u=3x^2+1\), giving \(du=6x\,dx\). Then:
Check by differentiation:
The derivative recovers the original integrand. The exponent \(-3/2\) is not \(-1\), so the power rule applies.
Worked Example: A Quadratic with a Linear Derivative Factor
Evaluate \(\int \frac{2x+1}{\sqrt{x^2+x+4}}\,dx\).
The inner expression is \(x^2+x+4\), and its derivative is \(2x+1\), exactly the numerator. With \(u=x^2+x+4\), we have \(du=(2x+1)\,dx\). Therefore:
Check the result:
The radicand is always positive: \(x^2+x+4=(x+\frac12)^2+\frac{15}{4}>0\). Thus the expression and its reciprocal square root are defined for every real \(x\).
When the Pattern Does Not Match
Substitution is useful here because the derivative factor is present. A radical by itself does not guarantee that this method applies directly. For example, in \(\int \sqrt{x^2+1}\,dx\), the derivative of the radicand is \(2x\), but there is no factor \(2x\) multiplying the radical. The substitution \(u=x^2+1\) would leave an \(x\) factor to express in terms of \(u\), so it does not immediately produce a power-rule integral.
This observation is not a claim that such an integral has no antiderivative. It means only that the direct derivative-match pattern in this tutorial is absent. Before substituting, identify the inner function, differentiate it, and compare that derivative with the rest of the integrand.
A Common Mistake: Losing a Constant
In the second example, a common error is to integrate \(u^{-1/2}\) as \(u^{1/2}\), forgetting to divide by the new exponent \(1/2\). That gives the incorrect claim:
Differentiate this proposed answer. The result is:
which is only half the original integrand. The correction is to divide by \(1/2\), or equivalently multiply by \(2\):
The derivative check is a reliable way to catch a missing coefficient. It verifies both the power and the constant produced by the substitution.
Why This Pattern Matters
Recognizing a composite radical and its derivative turns an integral in \(x\) into a basic power integral in \(u\). The important work is not manipulating the radical itself; it is identifying the inner expression and checking that its derivative is present, possibly after accounting for a constant.
For a quick decision, use this sequence:
- Identify the expression \(g(x)\) under the square root.
- Compute \(g'(x)\) and compare it with the factor outside the radical.
- Set \(u=g(x)\), replacing \(g'(x)\,dx\) by \(du\), including any constant multiplier.
- Rewrite the radical or reciprocal radical as a power of \(u\).
- Integrate with the power rule when the exponent is not \(-1\), substitute back, and check by differentiation.
Keeping these steps separate helps prevent two frequent errors: choosing an inner function whose derivative is not present, and losing a constant while applying the power rule. The final derivative check confirms that the proposed antiderivative reproduces the original integrand on the interval under consideration.
Check Your Understanding
For each question, identify the substitution pattern and pay attention to the exponent or constant factor.
- For \(\int 8x\sqrt{4x^2+3}\,dx\), what substitution makes the integral a power of \(u\), and what is the antiderivative?
- Evaluate \(\int \frac{10x}{\sqrt{5x^2+2}}\,dx\). What derivative check confirms your coefficient?
- Rewrite \(1/(\sqrt{g(x)})^5\) as a power of \(g(x)\).
- Why does the power rule stated here exclude exponent \(-1\)?
- Does the direct derivative-match pattern apply to \(\int \sqrt{x^2+4}\,dx\)? Explain by comparing the derivative of the radicand with the remaining factor.
