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When faced with certain integrals, students often jump straight to integration by parts because the formula is fresh in their minds or the integrand looks like a product. However, some questions that appear to call for integration by parts are actually far more straightforward when tackled with substitution. Recognising these cases can save you time and effort during exams!
A Classic Example
Let’s consider the following integral:
\int x \cos(x^2) \, dxAt first glance, you might see the product x \cdot \cos(x^2) and think, “This is perfect for integration by parts!” But if you try to set up integration by parts here, the resulting integrals quickly become complicated and unwieldy.
Why Substitution Works Better
Look closer at the argument of the cosine function: x^2. If we let u = x^2, then du = 2x\,dx so x\,dx = \frac{1}{2}du.
Substituting, we get:
\int x \cos(x^2) \, dx = \int \cos(u) \cdot \frac{1}{2} du = \frac{1}{2} \int \cos(u) duNow, integrating \cos(u) is simple:
\frac{1}{2} \int \cos(u)\ du = \frac{1}{2} \sin(u) + C = \frac{1}{2} \sin(x^2) + CThis solution is much quicker and neater than using integration by parts, which would involve repeatedly differentiating and integrating, only to end up needing substitution anyway.
Takeaway for Students
When approaching an integral, always pause to consider substitution, especially if the integrand contains a function and its derivative (or something close). In the example above, the presence of x alongside \cos(x^2) is a strong hint that substitution will simplify the process.
For more strategies and practice with integrals, resources like an A Level Maths Revision Course often provide a variety of examples to help you recognize the most efficient method for any given problem.
Conclusion
Efficiency in calculus isn’t just about knowing all the techniques—it’s about choosing the right one for the job. Always check for substitution before diving into integration by parts, and you’ll save yourself time and avoid unnecessary complications.
