Trigonometric Integrals and Trigonometric Substitution

Section 8.3: Trigonometric Integrals

Integrals Involving Products of Sines and Cosines

Integrals of the form $\int \sin^m x \cos^n x \, dx$ are common in calculus. The strategies for evaluating these depend on whether the exponents $m$ and $n$ are odd or even.

• Case 1: If $m$ is odd, write $m = 2k + 1$ and use the identity $\sin^2 x = 1 - \cos^2 x$ to reduce the power of sine.

• Case 2: If $n$ is odd, write $n = 2k + 1$ and use the identity $\cos^2 x = 1 - \sin^2 x$ to reduce the power of cosine.

• Case 3: If both $m$ and $n$ are even, use the half-angle identities:

  • $\sin^2 x = \frac{1 - \cos 2x}{2}$

  • $\cos^2 x = \frac{1 + \cos 2x}{2}$

  to reduce the integrand to lower powers of $\cos 2x$ or $\sin 2x$.

Worked Example: $\int \cos^2(3x) \, dx$

To integrate $\cos^2(3x)$, use the half-angle identity:

• $\cos^2(\alpha) = \frac{1 + \cos(2\alpha)}{2}$

So,

• $\cos^2(3x) = \frac{1 + \cos(6x)}{2}$

Integrate term by term:

• $\int \cos^2(3x) \, dx = \int \frac{1}{2} \, dx + \int \frac{\cos(6x)}{2} \, dx$

• $= \frac{x}{2} + \frac{1}{2} \int \cos(6x) \, dx$

• $= \frac{x}{2} + \frac{1}{2} \cdot \frac{\sin(6x)}{6} + C$

• $= \frac{x}{2} + \frac{\sin(6x)}{12} + C$

Reduction Formulas for Powers of Sine and Cosine

Reduction formulas allow us to express integrals of higher powers in terms of lower powers. For example, for $I_n = \int \cos^n(x) dx$:

• $I_n = \frac{1}{n} \cos^{n-1}(x) \sin(x) + \frac{n-1}{n} I_{n-2}$

These formulas are useful for systematically reducing the power of the trigonometric function until a basic integral is reached.

Integrals Involving Powers of Tangent and Secant

For integrals of the form $\int \tan^m(x) \sec^n(x) dx$:

• If $n$ is even, save a $\sec^2(x)$ factor and use $\sec^2(x) = 1 + \tan^2(x)$.

• If $m$ is odd, save a $\sec(x)\tan(x)$ factor and use $\tan^2(x) = \sec^2(x) - 1$.

Reduction formulas also exist for these integrals, allowing for systematic reduction.

Section 8.4: Trigonometric Substitutions

Introduction to Trigonometric Substitution

Trigonometric substitution is a technique for evaluating integrals involving square roots of quadratic expressions. The method replaces the variable with a trigonometric function, simplifying the radical.

• For $\sqrt{a^2 - x^2}$, use $x = a \sin \theta$

• For $\sqrt{a^2 + x^2}$, use $x = a \tan \theta$

• For $\sqrt{x^2 - a^2}$, use $x = a \sec \theta$

Procedure for Trigonometric Substitution

1. Write the substitution for $x$, calculate $dx$, and specify the range for $\theta$.

2. Substitute into the integral and simplify.

3. Integrate with respect to $\theta$.

4. Draw a reference triangle to convert back to $x$.

Reference Triangles and Inverse Trigonometric Functions

Reference triangles help convert the result back to the original variable. The limits and expressions for $\theta$ are related to the inverse trigonometric functions:

• $\theta = \tan^{-1}\left(\frac{x}{a}\right)$

• $\theta = \sin^{-1}\left(\frac{x}{a}\right)$

• $\theta = \sec^{-1}\left(\frac{x}{a}\right)$

Examples of Trigonometric Substitution

• Example: $\int \frac{dx}{\sqrt{9 - x^2}}$ Substitute $x = 3 \sin \theta$, $dx = 3 \cos \theta d\theta$, $\sqrt{9 - x^2} = 3 \cos \theta$. The integral becomes $\int d\theta = \theta + C = \sin^{-1}(x/3) + C$.

• Example: $\int \frac{dx}{x^2 + 4}$ Substitute $x = 2 \tan \theta$, $dx = 2 \sec^2 \theta d\theta$, $x^2 + 4 = 4 \sec^2 \theta$. The integral becomes $\int \frac{2 \sec^2 \theta}{4 \sec^2 \theta} d\theta = \frac{1}{2} \int d\theta = \frac{1}{2} \tan^{-1}(x/2) + C$.

Note: Always check if a simpler substitution is possible before using trigonometric substitution.

Summary Table: Trigonometric Substitutions

Radical	Substitution	Triangle Side
$\sqrt{a^2 - x^2}$	$x = a \sin \theta$	Opposite
$\sqrt{a^2 + x^2}$	$x = a \tan \theta$	Opposite
$\sqrt{x^2 - a^2}$	$x = a \sec \theta$	Hypotenuse

Additional info: The use of reference triangles and inverse trigonometric functions ensures that the substitution can be reversed and the answer expressed in terms of the original variable.