https://youtu.be/lpp6YLWB2GM

In this video I go over an example on trig substitution for integrals and solve for the integral of the function sqrt(9-x<sup>2</sup>)/x<sup>2</sup>. In this example I write the variable x in terms of a sine function to simplify the integral by removing the square root.

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# Trigonometric Substitution for Integrals: Example 1

![Trigonometric Substitution Example 1.jpeg](https://cdn.steemitimages.com/DQmZQsP26TK2xsjjHo4WEQ1TyFf58FGv777FrVybvt6vxxT/Trigonometric%20Substitution%20Example%201.jpeg)

**Example:**

![](https://cdn.steemitimages.com/DQmcUTTKaA9jXWJZbEq8NuG4H2hSY2SDHqBMFS6DAeV344C/image.png)

**Solution:**

![](https://cdn.steemitimages.com/DQmaBkV9cFMErzNzKstHszc9nTQBU4k5FyhDmvGpi2vR1RK/image.png)

![](https://cdn.steemitimages.com/DQmdJZ2j5ZuEZEqzXv7yiQkVHzrjj4KCrezNyCJtbFy7AP4/image.png)

Since this is an indefinite integral, we must return to the original variable x. This can be done by using trigonometric identities to express cotѲ in terms of sinѲ = x/3 or by drawing a diagram where Ѳ is considered as the angle of a right triangle. The latter is usually easier and we will try that:

![](https://cdn.steemitimages.com/DQmSw9Wvuapw4GCUkKUsETNH39io4JaN2jvifw37TXeHMAr/image.png)

Note: Although Ѳ > 0 in the diagram, this expression for cotѲ is valid even when Ѳ < 0.

![](https://cdn.steemitimages.com/DQmcgYsV3nPhb3XczgcyqX2nDnCJiaDiaqRV9rzhwLPuSgs/image.png)