notes/education/math/MATH1220 (calc II)/Integration with Trig Identities.md

The below integration makes use of the following trig identities:

1. The Pythagorean identity: \sin^2(x) + \cos^2(x) = 1
2. The derivative of sine: \frac{d}{dx}sin(x) = cos(x)
3. The derivative of cosine: \dfrac{d}{dx} \cos(x) = -\sin(x)
4. Half angle cosine identity: \cos^2(x) = \frac{1}{2}(1 + \cos(2x))
5. Half angle sine identity: \sin^2(x) = \frac{1}{2}(1 - \cos(2x))
6. tan^2(x) + 1 = sec^2(x)
7. \dfrac{d}{dx}(\tan(x)) = \sec^2(x) \Rightarrow \int \sec^2(x)dx = \tan(x) + C
8. \dfrac{d}{dx}(\sec x) = \sec(x)\tan(x) \Rightarrow \int\sec(x)\tan(x) dx = \sec(x) + C

Examples

Evaluate the integral \int\sin^5(x)dx

1. With trig identities, it's common to work backwards with u-sub. In the above example, we can convert the equation into simpler cosine functions by setting du to -\sin(x)dx. This means that u is equal to cos(x).

\int\sin^4(x)\sin(x)dx

2. Rewrite sin^4(x) to be (\sin^2(x))^2 to take advantage of the trig identity 1 - \cos^2(x) = \sin^2(x)

\int(\sin^2x)^2 \sin(x)dx

3. Apply the above trig identity and substitute u:

\int(1 - u^2)^2 (-du)

4. Foil out and move negative out of integral:

-\int(1 - 2u^2 + u^4)du

5. Take advantage of the distributive property of integrals:

- (u - \frac{2}{3}u^3 + \frac{1}{5}u^5) + C

6. Substituting \cos(x) back in for u, we get the evaluated (but not entirely simplified) integral:

-(\cos(x)- \frac{2}{3}\cos^3x + \frac{1}{5}\cos^5x)

Trigonometric Substitutions

Trigonometric substitution is useful for equations containing \sqrt{a^2 + x^2} or a^2 + x^2, where a is any constant. It removes any addition or subtraction.

The general process involves the use of a trig identity and multiplying everything in that identity by a constant.

Consider the identity:

1 + \tan^2(\theta) = \sec^2(\theta)

Multiplying both sides of the identity by a^2, we get:

a^2 + a^2\tan^2(\theta) = a^2\sec^2(\theta)

This enables us to make use of substitution to simplify many integrals.

• x = a\tan \theta
• dx = a \sec^2\theta d\theta
• for -\frac{\pi}{2} < \theta < \frac{\pi}{2}

Examples

Evaluate the integral \int\frac{3}{4+x^2}dx

1. Move the constant coefficient out of the integral:

\int \frac{3}{4+x^2}dx = 3\int \frac{1}{4 + x^2}dx

2. Let x = 2tan\theta and dx = (2sec^2\theta d\theta), substitute accordingly

= 3\int\frac{1}{4 + 4\tan^2\theta}(2\sec^2\theta)d\theta

3. Factor 4 out in the denominator

= 3\int\frac{1}{4(1 + \tan^2\theta)}(2\sec^2\theta)d\theta

4. Considering the identity 1 + \tan^2 \theta = \sec^2 \theta

= 3\int\frac{1}{4(\sec^2\theta)}(2\sec^2\theta)d\theta

5. \sec^2\theta is present in the numerator and the denominator, so we can cancel those out. This means that:

3\int\frac{2}{4}d\theta = \frac{3}{2} \theta + C

6. At this point, we want to determine what \theta is equal to relative to x.
   1. Look back to step 2 we defined x = 2\tan\theta
   2. Moving 2 to the other side to,