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How to Use Sine Substitution to Integrate

With the trigonometric substitution method, you can do integrals containing radicals of certain forms because they match up with trigonometric functions. A sine can take the place of a radical in a particular form.

image0.png
  1. Draw a right triangle where

    image1.png

    You should confirm this with the Pythagorean theorem.

    image2.png image3.jpg
  2. Solve

    image4.png

    differentiate, and solve for dx.

    image5.png
  3. Find which trig function equals the radical over the a, and then solve for the radical.

    Look at the triangle in the figure.

    image6.png
  4. Use the results from Steps 2 and 3 to make substitutions in the original problem and then integrate.

    Note that in this particular problem, you have to make three substitutions, not just two like in the first example.

    image7.png
  5. The triangle shows that

    image8.png

    Now, substitute back for your final answer.

    image9.png

It’s a walk in the park.

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