# Lecture #9

Today, we completed the discussion of trigonometric integrals, which are integrals with integrands being

• powers of cosine times powers of sine; or
• powers of tangent times powers of secant.

Other type of trigonometric integral may be converted to the above types, and cotangent / cosecant pairs follow similar rules. The scan is here: wk4day9.

The above are especially important because they come up in important integrals in applications involving

• $$\sqrt{a^2-x^2}$$
• $$\sqrt{a^2+x^2}$$
• $$\sqrt{x^2-a^2}$$

which arises when dealing with circles, ellipses and right-angled triangles, for instance.

The integration of function with the above terms will require a new type of substitution, which is an inverse substitution using trigonometric functions. This topic is called integration by trigonometric substitution and will not be tested in midterm 1.

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