Two Different Roles for Trigonometry
Trigonometric functions can enter an integration problem in two different ways. An integrand such as \(\sin^3 x\cos^2 x\) already consists of trigonometric functions of the variable. It is usually best to simplify those functions with identities, then integrate the resulting expression.
An integrand such as \(1/\sqrt{9-x^2}\), by contrast, contains an algebraic radical built from a quadratic expression in \(x\). A trigonometric substitution may simplify that radical because a fundamental identity can turn the quadratic expression into a square.
The key is to match the method to the structure. A trigonometric substitution is not a general-purpose way to integrate any expression containing sine or cosine, and a trigonometric identity does not usually remove a quadratic radical in \(x\) by itself.
Identities for Powers of Sine and Cosine
For products of powers of sine and cosine, the parity of the exponents can help identify a useful rewrite. In brief, an odd power of one function allows one factor to be set aside while the remaining even power is rewritten using \(\sin^2 x+\cos^2 x=1\). If both powers are even, a double-angle identity can reduce the powers. These are signals for using identities; the best choice still depends on the actual integrand.
For example, in \(\sin^3 x\cos^2 x\), one factor of \(\sin x\) can be set aside. Rewriting the remaining \(\sin^2 x\) in terms of \(\cos x\) leaves a factor that fits the differential \(d(\cos x)\). This is an ordinary substitution following an identity—not a trigonometric substitution of \(x\) itself.
Worked Example: An Odd Sine Power
Evaluate \(\int \sin^3 x\cos^2 x\,dx\). Separate one factor of \(\sin x\), and rewrite the other two using \(\sin^2 x=1-\cos^2 x\):
Let \(u=\cos x\), so \(du=-\sin x\,dx\). Then
To check, differentiate the result:
The derivative is the original integrand.
When both powers are even, a double-angle identity can be a more direct fit. In the next example, reducing the powers gives a simple expression in a constant and a single cosine.
Worked Example: Both Powers Even
Evaluate \(\int \sin^2 x\cos^2 x\,dx\). First use \(\sin x\cos x=\frac12\sin 2x\), then use the cosine double-angle identity:
Integrate the two terms:
For the check, differentiate the proposed antiderivative:
Again, the result differentiates to the original integrand.
When a Trigonometric Substitution Fits
A trigonometric substitution replaces an algebraic variable with a trigonometric expression chosen to exploit a trigonometric identity. Its purpose here is to simplify a radical involving a quadratic expression. For instance, if the radical is \(\sqrt{a^2-x^2}\), the substitution \(x=a\sin\theta\) makes the expression inside the square root a multiple of \(\cos^2\theta\).
The domain matters. Since a square root denotes the nonnegative root, replacing \(\sqrt{\cos^2\theta}\) by \(\cos\theta\) is valid only where \(\cos\theta\geq 0\). For \(x=a\sin\theta\), choosing \(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\) ensures this condition and makes the substitution one-to-one. A substitution should simplify the integral while respecting the sign and domain of the original expression.
Why the Radical Simplifies
For \(a>0\), set \(x=a\sin\theta\) with \(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\). Then \(dx=a\cos\theta\,d\theta\). The identity \(\sin^2\theta+\cos^2\theta=1\) gives
The last equality uses both \(a>0\) and \(\cos\theta>0\) on the chosen interval. The substitution is effective when this simpler radical and the corresponding \(dx\) turn the whole integrand into an easier expression in \(\theta\). The identity alone does not guarantee that the resulting integral is simpler, so inspect the complete transformed integrand.
Worked Example: A Quadratic Radical
Evaluate \(\int \frac{dx}{\sqrt{9-x^2}}\) for \(-3<x<3\). The radical suggests \(x=3\sin\theta\), with \(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\). Then
Substitute both expressions into the integral. Because \(\cos\theta>0\) on the chosen interval, the factors cancel:
In the last line, \(\sin\theta=x/3\) and the chosen range of \(\theta\) gives \(\theta=\arcsin(x/3)\). Check by differentiating, for \(-3<x<3\):
This matches the integrand on the stated domain.
A Practical Choice of Method
Before calculating, identify what kind of object the variable appears inside. The following comparison is a guide, not a substitute for checking whether a proposed rewrite actually simplifies the integral.
| Structure in the integrand | First method to consider | Reason |
|---|---|---|
| Powers or products of \(\sin x\) and \(\cos x\) | Trigonometric identities | Identities can reduce powers or expose a factor suited to an ordinary substitution. |
| A radical involving a quadratic expression in \(x\) | Consider a trigonometric substitution | A suitable substitution may convert the expression under the radical using a trigonometric identity. |
| Both types of structure | Simplify and compare the resulting forms | The best method depends on which parts of the integrand become simpler together. |
A method is useful when it reduces the work on the complete integrand. A substitution that simplifies one factor but makes the rest more complicated may not be an improvement. This is why structural recognition should be followed by a quick check of what the transformed integral will look like.
A Common Mistake: Confusing the Differentials
In the first worked example, setting \(u=\cos x\) gives \(du=-\sin x\,dx\), not \(du=-\cos^2 x\,dx\). A tempting but incorrect calculation would be
This incorrectly treats \(\cos^2 x\,dx\) as the differential factor. It also fails the derivative check: differentiating the proposed expression with \(u=\cos x\) gives
which is not \(\sin^3 x\cos^2 x\). The correction is to set aside \(\sin x\,dx\), since that is the factor that matches \(du=-\sin x\,dx\), and then rewrite the remaining powers in terms of \(u=\cos x\).
Why the Distinction Matters
Choosing between identities and trigonometric substitution is a matter of matching the algebra to its structure. Trigonometric powers are already functions of a trigonometric argument, so identities can reorganize them directly. A quadratic radical in \(x\) may instead call for a change of variable that makes the expression under its root easier to handle.
Either way, two checks help keep the method sound: confirm that the substitution or identity accounts for every factor, and differentiate the proposed antiderivative to see whether it returns the original integrand. For a trigonometric substitution involving a radical, also check the domain and the sign of the square root.
Check Your Understanding
Use these questions to test your choice of method and your attention to conditions.
- For \(\int \sin^5 x\cos^2 x\,dx\), which trigonometric factor could you set aside to match an ordinary substitution, and why?
- Which identity reduces \(\sin^2 x\cos^2 x\) to an expression involving \(\cos 4x\)?
- For \(x=3\sin\theta\) with \(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\), why is \(\sqrt{9-x^2}=3\cos\theta\), rather than \(-3\cos\theta\)?
- What structural feature of an integrand suggests considering a trigonometric substitution rather than applying identities to powers of sine and cosine?
- Why is differentiating an antiderivative a useful check on both the algebra and the method?
