Trigonometric Integrals

1. Explanatory Notes

When our basic integration tools are limited, evaluating integrals that contain powers or products of trigonometric functions requires rewriting the integrand into an integrable form. The primary strategy of Skill 1 relies on substituting high-power expressions with equivalent lower-power expressions using fundamental trigonometric identities.

Depending on whether the exponent is odd or even, you will choose between two major families of identities:

1. Explanatory Notes

When integrating products or powers of trigonometric functions, standard integration formulas cannot always be applied directly. Skill 2 focuses on strategically breaking apart an integrand to isolate a specific single or squared factor. This isolated factor is deliberately saved to serve as the differential ($du$) component for a future $u$-substitution.

The choice of which factor to split off depends on the relationship between the powers of the trigonometric functions involved:

1. Explanatory Notes

Evaluating trigonometric integrals with exceptionally high exponents (such as $\int \sin^{8}(x) \, dx$) using basic expansion or half-angle identities can become incredibly tedious and time-consuming. To ease this operational workload, mathematicians and engineers use Reduction Formulas.

A reduction formula is an integration rule derived using integration by parts that equates an integral involving a high power of a trigonometric function with an algebraic expression added to an alternative integral where the power has been reduced. By applying these formulas repeatedly, a high-degree expression can step-by-step be broken down until it reaches a low-degree power that can be integrated using basic calculus rules.

The standard reduction formulas for positive integers $n$ include:

• Sine Power Reduction:

  $$\int \sin^{n}(x) \, dx = -\frac{\sin^{n-1}(x)\cos(x)}{n} + \frac{n-1}{n}\int \sin^{n-2}(x) \, dx$$

• Cosine Power Reduction:

  $$\int \cos^{n}(x) \, dx = \frac{\cos^{n-1}(x)\sin(x)}{n} + \frac{n-1}{n}\int \cos^{n-2}(x) \, dx$$

• Tangent Power Reduction (where $n \neq 1$):

  $$\int \tan^{n}(x) \, dx = \frac{\tan^{n-1}(x)}{n-1} - \int \tan^{n-2}(x) \, dx$$

• Secant Power Reduction (where $n \neq 1$):

  $$\int \sec^{n}(x) \, dx = \frac{\sec^{n-2}(x)\tan(x)}{n-1} + \frac{n-2}{n-1}\int \sec^{n-2}(x) \, dx$$

2. Pure Math Sample Problems