3.3 Trigonometric Substitution

Trigonometric substitution is a powerful tool for tackling integrals with tricky square roots. It transforms complex expressions into more manageable trigonometric functions, making integration easier. This technique is especially handy when dealing with quadratic expressions under square roots.

Knowing when to use each substitution is key. For $\sqrt{a^2 - x^2}$, use $x = a\sin\theta$. For $\sqrt{a^2 + x^2}$, go with $x = a\tan\theta$. And for $\sqrt{x^2 - a^2}$, opt for $x = a\sec\theta$. Remember to convert your final answer back to x!

Trigonometric Substitution

Trigonometric substitution for square roots

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• Technique used to evaluate integrals containing square roots of quadratic expressions ($ax^2 + bx + c$, where $a \neq 0$)
• Transforms the integral into one involving trigonometric functions may be easier to evaluate
• After substitution, use trigonometric identities and standard integration techniques to solve the integral
• Three main forms of quadratic expressions suitable for trigonometric substitution:
  • $\sqrt{a^2 - x^2}$ (difference of squares)
  • $\sqrt{a^2 + x^2}$ (sum of squares)
  • $\sqrt{x^2 - a^2}$ (difference of squares with $x^2$ first)

Choosing appropriate substitutions

• For integrals containing $\sqrt{a^2 - x^2}$, use the substitution $x = a\sin\theta$
  • $dx = a\cos\theta d\theta$
  • $\sqrt{a^2 - x^2} = a\sqrt{1 - \sin^2\theta} = a\cos\theta$ (Pythagorean identity)
  • Example: $\int \sqrt{9 - x^2} dx$ with $a = 3$
• For integrals containing $\sqrt{a^2 + x^2}$, use the substitution $x = a\tan\theta$
  • $dx = a\sec^2\theta d\theta$
  • $\sqrt{a^2 + x^2} = a\sqrt{1 + \tan^2\theta} = a\sec\theta$ (Pythagorean identity)
  • Example: $\int \frac{x}{\sqrt{4 + x^2}} dx$ with $a = 2$
• For integrals containing $\sqrt{x^2 - a^2}$, use the substitution $x = a\sec\theta$
  • $dx = a\sec\theta\tan\theta d\theta$
  • $\sqrt{x^2 - a^2} = a\sqrt{\sec^2\theta - 1} = a\tan\theta$ (Pythagorean identity)
  • Example: $\int \sqrt{x^2 - 1} dx$ with $a = 1$

Converting solutions to x-expressions

• After evaluating the integral using trigonometric substitution, the solution will be in terms of $\theta$
• To express the solution in terms of the original variable $x$, use the inverse of the substitution made:
  1. For $x = a\sin\theta$, use $\theta = \arcsin(\frac{x}{a})$
  2. For $x = a\tan\theta$, use $\theta = \arctan(\frac{x}{a})$
  3. For $x = a\sec\theta$, use $\theta = \arcsec(\frac{x}{a})$
• Replace trigonometric functions of $\theta$ with their corresponding expressions in terms of $x$
• Simplify the resulting expression, if possible, to obtain the final solution in terms of $x$
• Example: If the solution is $2\theta + \sin\theta + C$ and the substitution was $x = 3\tan\theta$, the x-expression would be $2\arctan(\frac{x}{3}) + \frac{x}{\sqrt{9+x^2}} + C$

Related techniques and applications

• Integration by parts: A complementary method often used in conjunction with trigonometric substitution for more complex integrals
• Hyperbolic functions: Similar substitutions can be made using hyperbolic functions for certain types of integrals
• Partial fractions: Another technique that can be combined with trigonometric substitution to solve more complicated rational integrals
• Differential equations: Trigonometric substitution is sometimes employed in solving certain types of differential equations