Trigonometric Substitution
Use x = a sinθ for √(a²−x²).
Key idea
Use x = a sinθ for √(a²−x²).
In depth
For √(a² − x²), √(a² + x²), √(x² − a²), use x = a sin θ, a tan θ, a sec θ. The radical becomes a trig function, and the integral simplifies. Convert back with a right triangle.
Worked example
∫ dx/√(4 − x²): x = 2 sin θ gives ∫ dθ = θ = arcsin(x/2) + C.
Real-world uses
- Computing the arc length of a circle segment and integrals used in geometry and physics
Common mistake
Forgetting dx = a cos θ dθ.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Trigonometric Substitution?
Use x = a sinθ for √(a²−x²). It is a topic at the expert level in Calculus.
Where is Trigonometric Substitution used in real life?
Real-world examples: Computing the arc length of a circle segment and integrals used in geometry and physics.
What should I learn before Trigonometric Substitution?
Make sure you are comfortable with Trigonometric Integrals, the previous topic in Calculus.
What is a common mistake with Trigonometric Substitution?
Forgetting dx = a cos θ dθ.
How can I practise Trigonometric Substitution?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Trigonometric Substitution?
Move on to Partial Fractions Integration next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.