Trigonometric Integrals Lecture Notes

Practice flashcards covering the integration strategies for trigonometric functions including sine, cosine, tangent, and secant as discussed in the lecture.

Strategy for Odd Powers of Sine ($k$ is odd)

Reduce the power of sine by one, convert the remaining even powers of sine into cosine using the identity $\sin^2(x) = 1 - \cos^2(x)$, and then perform a u-substitution with $u = \cos(x)$.

Strategy for Odd Powers of Cosine ($j$ is odd)

Reduce the power of cosine by one, convert the remaining even powers of cosine into sine using the identity $\cos^2(x) = 1 - \sin^2(x)$, and then perform a u-substitution with $u = \sin(x)$.

Sine Double Angle Formula

$$\sin^2(x) = \frac{1 - \cos(2x)}{2}$$

Cosine Double Angle Formula

$$\cos^2(x) = \frac{1 + \cos(2x)}{2}$$

Anti-derivative of $\tan(x)$

$$\ln|\sec(x)| + C$$

Anti-derivative of $\sec(x)$

$$\ln|\sec(x) + \tan(x)| + C$$

Anti-derivative of $\sec^2(x)$

$$\tan(x) + C$$

Secant-Tangent Strategy (Secant power $j$ is even)

Choose $u = \tan(x)$, peel off a $\sec^2(x)$ factor to serve as the derivative ($du$), and convert any remaining even powers of secant into tangents using the identity $\sec^2(x) = 1 + \tan^2(x)$.

Secant-Tangent Strategy (Tangent power $k$ is odd)

Choose $u = \sec(x)$, peel off a $\sec(x)\tan(x)$ factor to serve as the derivative ($du$), and convert the remaining even powers of tangent into secants.

Pure Tangent Power Strategy ($j = 0$)

Separate the integral into $\tan^2(x)$ times the remaining tangents, rewrite $\tan^2(x)$ as $\sec^2(x) - 1$, and distribute to reduce the power.

Pure Secant Power Strategy ($k = 0, j$ is odd)

Utilize integration by parts, often separating the function into $\sec(x)$ and $\sec^2(x)$ to find the anti-derivative.

Integration of $\sec^3(x)$ outcome

$$\int \sec^3(x)\,dx = \frac{1}{2}(\sec(x)\tan(x) + \ln|\sec(x) + \tan(x)|) + C$$

Product of Sine and Cosine with Scaling Factors

Requires the use of sum and difference formulas to convert products like $\sin(Ax)\cos(Bx)$ into sums of trigonometric functions.