Trigonometric Integrals

1. Basic Trigonometric Identities:

  • Sine and Cosine Identities:
    • sin2(x)+cos2(x)=1
    • tan(x)=sin(x)cos(x)
    • sec(x)=1cos(x), csc(x)=1sin(x)

2. Trigonometric Integrals:

  • Common Integrals:
    • sin(x)dx=cos(x)+C
    • cos(x)dx=sin(x)+C
    • tan(x)dx=lncos(x)+C
    • sec(x)dx=lnsec(x)+tan(x)+C
    • csc(x)dx=lncsc(x)+cot(x)+C

3. Trigonometric Substitution:

  • Principle: Substitutes trigonometric expressions to simplify integrals involving radicals.
  • Substitutions:
    • For a2x2, use x=asin(θ)
  • For x2a2, use x=asec(θ)
  • For x2+a2, use x=atan(θ)
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4. Trigonometric Identities and Techniques:

  • Double Angle Formulas:

    • sin(2x)=2sin(x)cos(x)
    • cos(2x)=cos2(x)sin2(x)
    • tan(2x)=2tan(x)1tan2(x)
  • Half-Angle Formulas:

    • sin2(x)=1cos(2x)2
    • cos2(x)=1+cos(2x)2

Types of Trigonometric Integrals:

  1. Basic Trigonometric Integrals:

    • ∫ sin(x) dx = -cos(x) + C
    • ∫ cos(x) dx = sin(x) + C
    • These are fundamental integrals involving sine and cosine functions.
  2. Powers of Trigonometric Functions:

    • Sinn(x) Cosm(x) dx
    • Integrating powers of sine and cosine functions involves various techniques like trigonometric identities or reduction formulas.
  3. Products of Trigonometric Functions:

    • Integrals of products of trigonometric functions like sin(x)cos(x) or sin(x)sin(2x) can be solved using substitution or integration by parts.
  4. Trigonometric Substitution:

    • Substituting trigonometric expressions to simplify integrals involving radicals or square roots.
    • Common substitutions include letting x=asin(θ) or x=atan(θ) to simplify the integral.

Tips for Solving Trigonometric Integrals:

  1. Trigonometric Identities:

    • Utilize trigonometric identities to simplify integrals involving trigonometric functions.
    • Examples include double-angle, half-angle, and Pythagorean identities.
  2. Symmetry Properties:

    • Exploit the symmetry properties of trigonometric functions (even or odd) to simplify integrals.
    • Utilize periodicity properties to reduce the range of integration.
  3. Integration by Parts:

    • Helpful when dealing with products of trigonometric functions or functions that don't lend themselves to straightforward substitutions.
    • Use the formula ∫ u dv = uv - ∫ v du.
  4. Trigonometric Reduction Formulas:

    • Useful for reducing powers of trigonometric functions into simpler expressions.
    • Example: The reduction formula ∫ Sinn(x) dx = -1/n * Sin(n-1)(x)cos(x) + (n-1)/n * ∫ Sin(n-2)(x) dx.

Example:

Let's solve the indefinite integral ∫ Sin2(x) dx:

Solution:

Using the trigonometric identity sin2(x)=1cos(2x)2, the integral becomes:

sin2(x)dx=12(1cos(2x))dx

12(x12sin(2x))+C

Example:

Let's solve the indefinite integral ∫ sin(x) cos(x) dx:

Solution:

To solve this integral, we'll use integration by parts. The formula for integration by parts is ∫ u dv = uv - ∫ v du.

Here, let's choose:

  • u=sin(x) (so du=cos(x)dx)
  • dv=cos(x)dx (so v=sin(x))

Applying the integration by parts formula: sin(x)cos(x)dx=sin(x)sin(x)sin(x)cos(x)dx

Let's rearrange this equation to solve for the integral on the left side: sin(x)cos(x)dx=sin2(x)sin(x)cos(x)dx

Now, move the integral term to the right side: 2sin(x)cos(x)dx=sin2(x) sin(x)cos(x)dx=sin2(x)2+C