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.