Calculus Volume 2 · free preview
Trigonometric Substitution
Roots of a squared minus x squared, a squared plus x squared, and x squared minus a squared convert by x equals a trig function into trigonometric integrals the previous section resolves, reversing the inverse trigonometric derivative formulas with full generality.
1. The three substitutions
The three substitutions
2. A worked example each form
A worked example each form
3. When the power is inside
When the power is inside
4. The reference triangle
The reference triangle
5. Root form choice
The radical sqrt(a^2 - x^2) is cleared by which substitution?
6. Sum form choice
The radical sqrt(a^2 + x^2) is cleared by which substitution?
7. Reference triangle purpose
After integrating in theta, what is the reference triangle's role?
8. Completing the square first
The integral of dx/sqrt(x^2 + 2x + 5) requires which preliminary step?
9. Gateway formula recovery
Applying x = a tan theta to the integral of dx/(a^2 + x^2) recovers which gateway formula?
10. Sine substitution differential
With x = a sin theta, the differential dx equals a cos theta d theta. What is the coefficient of cos theta d theta? Answer with a single letter.
11. Difference variable-first
The radical sqrt(x^2 - a^2) is cleared by which substitution, and what does the root become?
12. Root after tangent substitution
With x = a tan theta, the root sqrt(a^2 + x^2) becomes a times which function? Answer with the function name like sin theta.
13. Definite limit transformation
In a definite integral with x = 2 sin theta, the limit x = 1 becomes theta = pi/6. What does x = 0 become? Answer like pi/6.
14. Quadratic inside power
The form (x^2 + a^2)^(3/2) converts under x = a tan theta to a^3 times which power? Answer like cos cubed theta.
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