Calculus learning

Trigonometric Substitution: Choosing a Substitution and Its Domain

Trigonometric substitution replaces a quadratic radical with a trigonometric identity. Start with the sign of the squared terms, not a memorized triangle. A simple u-substitution may be shorter if the numerator already contains the derivative of the radicand.

Direct answer

Use x=a sin θ for √(a²−x²), x=a tan θ for √(a²+x²), or x=a sec θ for √(x²−a²).

Assume a>0 and choose an angle interval on which the substitution is one-to-one. Square roots are nonnegative; √(u²)=|u|.

Trigonometric Substitution worked method

For 1/√(a²−x²), set x=a sin θ with −π/2<θ<π/2. Then dx=a cos θ dθ and the positive cosine cancels the denominator.

The transformed integral is ∫1 dθ=θ+C. Return to x with θ=arcsin(x/a); differentiating confirms the result on |x|<a.

For a²+x², choose x=a tan θ with −π/2<θ<π/2. The identity 1+tan²θ=sec²θ removes the radical with sec θ>0.

For x²−a² on x>a, choose x=a sec θ with 0<θ<π/2. The negative branch x<−a needs its own angle/sign treatment; do not silently copy the positive branch.

For definite integrals, either change both bounds to θ immediately or return the antiderivative to x before inserting x-bounds. Mixing these two choices changes the problem.

1
∫ dx/√(a²−x²) = arcsin(x/a)+C
2
∫xa2+x2 dx=a2+x2+C\int \frac{x}{\sqrt{a^{2} + x^{2}}}\,\mathrm{d}x = \sqrt{a^{2} + x^{2}} + C
3
∫ dx/(a²+x²) = arctan(x/a)/a+C

Worked examples

∫ dx/√(16−x²)
arcsin(x/4)+C

The interval is −4<x<4.

∫x9+x2 dx\int \frac{x}{\sqrt{9 + x^{2}}}\,\mathrm{d}x
9+x2+C\sqrt{9 + x^{2}} + C

u=9+x² is shorter here.

∫ dx/(4+x²)
arctan⁡(x2)2+C\frac{\arctan\left(\frac{x}{2}\right)}{2} + C

The denominator has no real zero.

Common mistakes

  • • Dropping the absolute value before selecting an angle interval.
  • • Changing dx but not the denominator.
  • • Using x-bounds in a θ-integral.

Try the supported calculator preset

The preset opens a supported expression with its variable and conditions. Teaching examples elsewhere in this guide are reference explanations, not claims that the engine supports every method.

This runnable example is a quadratic rational integral. The radical derivations above are teaching references, not claims that the calculator implements every trig-substitution step.

Integrals:14+x2\frac{1}{4 + x^{2}}
Open the calculator without a preset: Integrals

References

Frequently asked questions

When should I try ordinary substitution first?+

When the remaining factors supply the derivative of the expression under the square root.

Does every radical require a triangle?+

No. Inverse-trigonometric identities can restore x directly.

Will the calculator show this method for every radical?+

No. Its rule set is narrower than this lesson. A limitation response is not a proof that the integral is impossible.

Conclusion

Trigonometric Substitution starts with the stated domain and conditions. Recheck those before carrying a worked example into a different problem.

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