1. Let's start by stating the problem: understanding how to solve trigonometric integrals. 2. Trigonometric integrals involve integrating functions like $\sin x$, $\cos x$, $\tan x$, and their powers or products. 3. A common formula used is the integral of sine and cosine: $$\int \sin x \, dx = -\cos x + C$$ $$\int \cos x \, dx = \sin x + C$$ where $C$ is the constant of integration. 4. Important rules: - Use substitution when you see a function and its derivative inside the integral. - Use identities like $\sin^2 x + \cos^2 x = 1$ to simplify powers. - For products like $\sin^m x \cos^n x$, use power-reducing or product-to-sum formulas. 5. Example: Integrate $\int \sin^2 x \, dx$. 6. Use the identity: $$\sin^2 x = \frac{1 - \cos 2x}{2}$$ 7. Substitute into the integral: $$\int \sin^2 x \, dx = \int \frac{1 - \cos 2x}{2} \, dx = \frac{1}{2} \int (1 - \cos 2x) \, dx$$ 8. Integrate term by term: $$\frac{1}{2} \int 1 \, dx - \frac{1}{2} \int \cos 2x \, dx = \frac{1}{2} x - \frac{1}{2} \cdot \frac{\sin 2x}{2} + C$$ 9. Simplify: $$\frac{1}{2} x - \frac{\sin 2x}{4} + C$$ 10. So the integral of $\sin^2 x$ is: $$\int \sin^2 x \, dx = \frac{1}{2} x - \frac{\sin 2x}{4} + C$$ This method applies similarly to other trigonometric integrals by using identities and substitution.