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Trigonometric substitution - Integration Techniques

Trigonometric substitution

In this section, we will look at evaluating trigonometric functions with trigonometric substitution. A lot of people normally substitute using trig identities, which you will have to memorize. However, Dennis will use a different and easier approach. He will use the idea of Pythagoras Theorem and make a relation with sine and cosine in terms of x to substitute. In more advance questions, we will use Pythagoras Theorem to make a relation with tangent and secant to substitute. No matter what type of questions you do, you will only need to know Pythagoras Theorem and nothing else!


Pre-requisite: Trigonometry Ratio: “SOH-CAH-TOA”
    • a)
      9x2xdx \int \sqrt{9-x^2}xdx
    • b)
      9x2dx \int \sqrt{9-x^2}dx
  • 2.
    Evaluate the integral (Type A: a2x2 \sqrt{a^2-x^2} ).
    • b)
      49x2x2dx \int \frac{\sqrt{49-x^2}}{x^2}dx
  • 3.
    Evaluate the integral (Type B: a2+x2 \sqrt{a^2+x^2} ).
    • b)
      x3(16x2+25)32dx \int \frac{x^3}{(16x^2+25)^{\frac{3}{2}}}dx
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Trigonometric substitution

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