Integration techniques such as trigonometric substitution can simplify integrals from "summary" of Skills in Mathematics - Integral Calculus for JEE Main and Advanced by Amit M Agarwal
Trigonometric substitution is a powerful technique that can be used to simplify integrals involving square roots, especially those in the form of $\sqrt{a^2 - x^2}$ or $\sqrt{a^2 + x^2}$. By making a suitable substitution involving trigonometric functions, we can transform these complicated expressions into more manageable trigonometric forms. This simplification allows us to apply standard trigonometric identities and formulas to evaluate the integral.
The key idea behind trigonometric substitution is to express the variable $x$ in terms of a trigonometric function such as sine, cosine, or tangent. Depending on the form of the square root expression, we choose the appropriate trigonometric substitution that will help us simplify the integral. For instance, if we have $\sqrt{a^2 - x^2}$, we can let $x = a\sin{\theta}$, which transforms the expression into $a\cos{\theta}$, making it easier to integrate.
Once we have made the trigonometric substitution, we can use trigonometric identities to simplify the integrand further. This typically involves expressing all terms in the integral in terms of the same trigonometric function, which allows us to combine terms and evaluate the int...
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