1. Let's start by stating the problem: Find the integral of trigonometric ratios such as $\sin x$, $\cos x$, $\tan x$, $\cot x$, $\sec x$, and $\csc x$. 2. The general formula for integration is to find a function whose derivative is the integrand. For trigonometric functions, we use known antiderivatives. 3. Important rules: - $\int \sin x \, dx = -\cos x + C$ - $\int \cos x \, dx = \sin x + C$ - $\int \tan x \, dx = -\ln|\cos x| + C$ - $\int \cot x \, dx = \ln|\sin x| + C$ - $\int \sec x \, dx = \ln|\sec x + \tan x| + C$ - $\int \csc x \, dx = -\ln|\csc x + \cot x| + C$ 4. Explanation: - The integral of $\sin x$ is $-\cos x$ because the derivative of $-\cos x$ is $\sin x$. - The integral of $\cos x$ is $\sin x$ because the derivative of $\sin x$ is $\cos x$. - For $\tan x$, rewrite as $\frac{\sin x}{\cos x}$ and use substitution $u=\cos x$. - Similarly for $\cot x$, rewrite as $\frac{\cos x}{\sin x}$ and use substitution $u=\sin x$. - The integrals of $\sec x$ and $\csc x$ involve multiplying numerator and denominator by expressions to facilitate substitution. 5. These integrals are fundamental in calculus and often used in solving differential equations and evaluating areas under curves involving trigonometric functions. Final answers: $$\int \sin x \, dx = -\cos x + C$$ $$\int \cos x \, dx = \sin x + C$$ $$\int \tan x \, dx = -\ln|\cos x| + C$$ $$\int \cot x \, dx = \ln|\sin x| + C$$ $$\int \sec x \, dx = \ln|\sec x + \tan x| + C$$ $$\int \csc x \, dx = -\ln|\csc x + \cot x| + C$$