Trig Integrals: Step-by-Step Guide
Standard trig integrals include: integral of sin x dx = -cos x + C, integral of cos x dx = sin x + C, integral of sec^2 x dx = tan x + C, integral of csc^2 x dx = -cot x + C, integral of sec x tan x dx = sec x + C, integral of csc x cot x dx = -csc x + C. More complex integrals use u-substitution, identities, or integration by parts.
Math Racer skill guide · Calculus 2 · AP Calc BC / College
How to solve it, step by step
- Recognize the integrand form. Match the integrand to a standard trig integral or identify a substitution.\[\int \sin x\,dx,\;\int \cos x\,dx,\;\int \sec^2 x\,dx\]
- Apply identities if needed. Use power-reduction, double-angle, or Pythagorean identities to rewrite into integrable pieces.\[\sin^2 x = \tfrac{1 - \cos 2x}{2}\]
- Use u-substitution if there is an inner function. Let u equal the inner expression and du equal its derivative.\[u = g(x),\;du = g'(x)\,dx\]
- Integrate term by term. Apply each standard antiderivative.\[\int \cos u\,du = \sin u + C\]
- Add the constant of integration and verify. Differentiate the result to confirm it returns the original integrand.\[\frac{d}{dx}F(x) = f(x)\]
Worked example
Evaluate integral of sin^2(x) dx.
Step 1: Not a standard form directly.
Step 2: Use power-reduction: sin^2 x = (1 - cos 2x)/2.
Step 3: Integral becomes integral of (1/2 - (1/2) cos 2x) dx.
Step 4: integral 1/2 dx = x/2. For integral (1/2) cos 2x dx, substitute u = 2x, du = 2 dx: (1/2) * (1/2) sin u = (1/4) sin 2x.
Step 5: Combine: x/2 - (1/4) sin 2x + C. Verify: d/dx [x/2 - (sin 2x)/4] = 1/2 - (2 cos 2x)/4 = 1/2 - cos(2x)/2 = (1 - cos 2x)/2 = sin^2 x. Confirmed.
Answer: integral of sin^2 x dx = x/2 - (sin 2x)/4 + C.
Tips and common mistakes
- Memorize the six standard trig antiderivatives - they appear constantly.
- Power reduction is essential for integrating sin^2, cos^2, sin^4, etc.
- For integral of sin^m x cos^n x dx with odd m: split off one sin and use cos^2 = 1 - sin^2.
- For integral of tan^n x dx: use tan^2 = sec^2 - 1 to peel off pairs.
- integral of sec x dx = ln|sec x + tan x| + C - non-obvious; memorize.
- integral of tan x dx = -ln|cos x| + C = ln|sec x| + C.
- Always check by differentiating your antiderivative.
- Don't forget the +C for indefinite integrals; for definite integrals, evaluate at the bounds.