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Integrals by Trigonometric Substitution
The document explains how to solve integrals using trigonometric substitution, specifically focusing on the integral dx/√(1 - x²). It outlines the steps for substitution where x = sin(t), leading to simplifications using trigonometric identities. The document further illustrates the principles behind trigonometric substitution for various forms of expressions including a² - x², a² + x², and x² - a².
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Integrals by Trigonometric Substitution
- 3. Integrals by TrigonometricSubstitution Let’s say we want to find this integral:
- 4. Integrals by TrigonometricSubstitution Let’s say we want to find this integral: dx √ 1 − x2
- 5. Integrals by TrigonometricSubstitution Let’s say we want to find this integral: dx √ 1 − x2 1. The denominator looks like a trig identity:
- 6. Integrals by TrigonometricSubstitution Let’s say we want to find this integral: dx √ 1 − x2 1. The denominator looks like a trig identity: cos2 t = 1 − sin2 t
- 7. Integrals by TrigonometricSubstitution Let’s say we want to find this integral: dx √ 1 − x2 1. The denominator looks like a trig identity: cos2 t = 1 − sin2 t We make the substitution:
- 8. Integrals by TrigonometricSubstitution Let’s say we want to find this integral: dx √ 1 − x2 1. The denominator looks like a trig identity: cos2 t = 1 − sin2 t We make the substitution: x2 = sin2 t
- 9. Integrals by TrigonometricSubstitution Let’s say we want to find this integral: dx √ 1 − x2 1. The denominator looks like a trig identity: cos2 t = 1 − sin2 t We make the substitution: x2 = sin2 t ⇒ x = sin t
- 10. Integrals by TrigonometricSubstitution Let’s say we want to find this integral: dx √ 1 − x2 1. The denominator looks like a trig identity: cos2 t = 1 − sin2 t We make the substitution: x2 = sin2 t ⇒ x = sin t We need to find dx:
- 11. Integrals by TrigonometricSubstitution Let’s say we want to find this integral: dx √ 1 − x2 1. The denominator looks like a trig identity: cos2 t = 1 − sin2 t We make the substitution: x2 = sin2 t ⇒ x = sin t We need to find dx: dx dt = d dt (sin t)
- 12. Integrals by TrigonometricSubstitution Let’s say we want to find this integral: dx √ 1 − x2 1. The denominator looks like a trig identity: cos2 t = 1 − sin2 t We make the substitution: x2 = sin2 t ⇒ x = sin t We need to find dx: dx dt = d dt (sin t) = cos t
- 13. Integrals by TrigonometricSubstitution Let’s say we want to find this integral: dx √ 1 − x2 1. The denominator looks like a trig identity: cos2 t = 1 − sin2 t We make the substitution: x2 = sin2 t ⇒ x = sin t We need to find dx: dx dt = d dt (sin t) = cos t dx = cos tdt
- 14. Integrals by TrigonometricSubstitution So, we can make the substitution:
- 15. Integrals by TrigonometricSubstitution So, we can make the substitution: x = sin t dx = cos tdt
- 16. Integrals by TrigonometricSubstitution So, we can make the substitution: x = sin t dx = cos tdt dx √ 1 − x2
- 17. Integrals by TrigonometricSubstitution So, we can make the substitution: x = sin t dx = cos tdt dx √ 1 − x2 dx = cos tdt 1 − sin2 t
- 18. Integrals by TrigonometricSubstitution So, we can make the substitution: x = sin t dx = cos tdt dx √ 1 − x2 dx = cos tdt :cos t 1 − sin2 t
- 19. Integrals by TrigonometricSubstitution So, we can make the substitution: x = sin t dx = cos tdt dx √ 1 − x2 dx = cos tdt :cos t 1 − sin2 t = cos t cos t dt
- 20. Integrals by TrigonometricSubstitution So, we can make the substitution: x = sin t dx = cos tdt dx √ 1 − x2 dx = cos tdt :cos t 1 − sin2 t = cos t cos t dt =
- 21. Integrals by TrigonometricSubstitution So, we can make the substitution: x = sin t dx = cos tdt dx √ 1 − x2 dx = cos tdt :cos t 1 − sin2 t = cos t cos t dt = dt
- 22. Integrals by TrigonometricSubstitution So, we can make the substitution: x = sin t dx = cos tdt dx √ 1 − x2 dx = cos tdt :cos t 1 − sin2 t = cos t cos t dt = dt = t + C
- 23. Integrals by TrigonometricSubstitution So, we can make the substitution: x = sin t dx = cos tdt dx √ 1 − x2 dx = cos tdt :cos t 1 − sin2 t = cos t cos t dt = dt = t + C Now we need to substitute back:
- 24. Integrals by TrigonometricSubstitution So, we can make the substitution: x = sin t dx = cos tdt dx √ 1 − x2 dx = cos tdt :cos t 1 − sin2 t = cos t cos t dt = dt = t + C Now we need to substitute back: x = sin t
- 25. Integrals by TrigonometricSubstitution So, we can make the substitution: x = sin t dx = cos tdt dx √ 1 − x2 dx = cos tdt :cos t 1 − sin2 t = cos t cos t dt = dt = t + C Now we need to substitute back: x = sin t ⇒ t = arcsin x
- 26. Integrals by TrigonometricSubstitution So, we can make the substitution: x = sin t dx = cos tdt dx √ 1 − x2 dx = cos tdt :cos t 1 − sin2 t = cos t cos t dt = dt = t + C Now we need to substitute back: x = sin t ⇒ t = arcsin x So:
- 27. Integrals by TrigonometricSubstitution So, we can make the substitution: x = sin t dx = cos tdt dx √ 1 − x2 dx = cos tdt :cos t 1 − sin2 t = cos t cos t dt = dt = t + C Now we need to substitute back: x = sin t ⇒ t = arcsin x So: dx √ 1 − x2 dx = arcsin x + C
- 28. Integrals by TrigonometricSubstitution So, we can make the substitution: x = sin t dx = cos tdt dx √ 1 − x2 dx = cos tdt :cos t 1 − sin2 t = cos t cos t dt = dt = t + C Now we need to substitute back: x = sin t ⇒ t = arcsin x So: dx √ 1 − x2 dx = arcsin x + C
- 29.
- 30. Why Trigonometric SubstitutionWorks Trigonometric substitution allows us to simplify radicals, because of the identities:
- 31. Why Trigonometric SubstitutionWorks Trigonometric substitution allows us to simplify radicals, because of the identities: 1 − cos2 t = sin2 t
- 32. Why Trigonometric SubstitutionWorks Trigonometric substitution allows us to simplify radicals, because of the identities: 1 − cos2 t = sin2 t 1 + tan2 t = sec2 t
- 33. Why Trigonometric SubstitutionWorks Trigonometric substitution allows us to simplify radicals, because of the identities: 1 − cos2 t = sin2 t 1 + tan2 t = sec2 t When you have an expression of the form:
- 34. Why Trigonometric SubstitutionWorks Trigonometric substitution allows us to simplify radicals, because of the identities: 1 − cos2 t = sin2 t 1 + tan2 t = sec2 t When you have an expression of the form: a2 − x2 ⇒ Use x = a sin t
- 35. Why Trigonometric SubstitutionWorks Trigonometric substitution allows us to simplify radicals, because of the identities: 1 − cos2 t = sin2 t 1 + tan2 t = sec2 t When you have an expression of the form: a2 − x2 ⇒ Use x = a sin t And you’ll get: a2 − x2 = a2 (1 − sin2 t) = a2 cos2 t
- 36. Why Trigonometric SubstitutionWorks Trigonometric substitution allows us to simplify radicals, because of the identities: 1 − cos2 t = sin2 t 1 + tan2 t = sec2 t When you have an expression of the form: a2 − x2 ⇒ Use x = a sin t And you’ll get: a2 − x2 = a2 (1 − sin2 t) = a2 cos2 t a2 + x2 ⇒ Use x = a tan t
- 37. Why Trigonometric SubstitutionWorks Trigonometric substitution allows us to simplify radicals, because of the identities: 1 − cos2 t = sin2 t 1 + tan2 t = sec2 t When you have an expression of the form: a2 − x2 ⇒ Use x = a sin t And you’ll get: a2 − x2 = a2 (1 − sin2 t) = a2 cos2 t a2 + x2 ⇒ Use x = a tan t And you’ll get: a2 + x2 = a2 (1 + tan2 t) = a2 sec2 t
- 38. Why Trigonometric SubstitutionWorks Trigonometric substitution allows us to simplify radicals, because of the identities: 1 − cos2 t = sin2 t 1 + tan2 t = sec2 t When you have an expression of the form: a2 − x2 ⇒ Use x = a sin t And you’ll get: a2 − x2 = a2 (1 − sin2 t) = a2 cos2 t a2 + x2 ⇒ Use x = a tan t And you’ll get: a2 + x2 = a2 (1 + tan2 t) = a2 sec2 t x2 − a2 ⇒ Use x = a sec t
- 39. Why Trigonometric SubstitutionWorks Trigonometric substitution allows us to simplify radicals, because of the identities: 1 − cos2 t = sin2 t 1 + tan2 t = sec2 t When you have an expression of the form: a2 − x2 ⇒ Use x = a sin t And you’ll get: a2 − x2 = a2 (1 − sin2 t) = a2 cos2 t a2 + x2 ⇒ Use x = a tan t And you’ll get: a2 + x2 = a2 (1 + tan2 t) = a2 sec2 t x2 − a2 ⇒ Use x = a sec t And you’ll get: x2 − a2 = a2 (sec2 t − 1) = a2 tan2 t