Basic Trigonometric Integration

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1) Identify the trig function

For a basic trigonometric integral, first recognize which standard trig function appears: (\sin x), (\cos x), (\tan x), (\sec x), and related forms. The main goal is to use the standard antiderivative and then simplify the result.

2) Use the matching antiderivative

Common examples include:

(\int \sin x,dx = -\cos x + C)
(\int \cos x,dx = \sin x + C)
(\int \sec^2 x,dx = \tan x + C)
(\int \csc^2 x,dx = -\cot x + C)
(\int \sec x\tan x,dx = \sec x + C)
(\int \csc x\cot x,dx = -\csc x + C)

If a constant multiple is present, factor it out first.

3) Apply the constant rule and simplify

Treat numerical factors separately: (\int kf(x),dx = k\int f(x),dx). After integrating, simplify signs and constants carefully. Keep the constant of integration (C) unless a definite integral is specifically given.

4) Check your answer

Differentiate your result to see whether you recover the original integrand. This is the best quick check for trig integrals, especially for sign errors.

5) Common mistakes to avoid

Watch for:

Missing a minus sign in integrals of (\sin x), (\csc^2 x), or (\csc x\cot x)
Forgetting to factor out constants
Leaving the answer unsimplified

A correct final answer should be exact, simplified, and consistent with the original trig function.