# Integration using trigonometric identities

### Integration using trigonometric identities

In this section, we will take a look at several methods for integrating trigonometric functions. All methods require us to use U-Substitution and substituting with trigonometric identities. In addition, the trigonometric functions we are dealing with are products of sine and cosine with powers. There are a total of 3 cases. The first case is when the power of cosine is odd. The second case is when the power of sine is odd. Lastly, the third case is when both the powers of sine and cosine are even.

#### Lessons

Pre-requisite: * Trigonometry –“Trigonometric Identities”

Note: Strategy for evaluating $\int \sin^{\blacksquare}x$ $\cos^{\blacksquare}x$ $dx$

Case 1: $\int \sin^{\blacksquare}x$ $\cos^{odd}x$ $dx$
1. strip out one $cosine$ factor
2. express the remaining $cosine$ factors in terms of $sine$ using the Pythagorean Identity: $\cos^2x=1-\sin^2x$
3. substitute $u=\sin x$

Case 2: $\int \sin^{odd}x$ $\cos^{\blacksquare}x$ $dx$
1. strip out one $sine$ factor
2. express the remaining $sine$ factors in terms of $cosine$ using the Pythagorean Identity: $\sin^2x=1-\cos^2x$
3. substitute $u=\cos x$

Case 3: $\int \sin^{even}x$ $\cos^{even}x$ $dx$
1. use the half-angle identities: $\sin^2x=\frac{1}{2}(1-\cos2x)$ $or$ $\cos^2x=\frac{1}{2}(1+\cos2x)$
2. if necessary, use the double-angle identity: $\sin x$ $\cos x$ $=\frac{1}{2}\sin2x$

• 1.
Evaluate the integral (odd power of cosine).
a)
$\int \sin^8x$ $\cos^3 x$ $dx$

b)
$\int \cos^5x$ $dx$

• 2.
Evaluate the integral (odd power of sine).
a)
$\int \sin^5x$ $\cos^2 x$ $dx$

b)
$\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \sin^5x$ $\cos^3 x$ $dx$

c)
$\int \sin^3x$ $\sqrt{\cos x}$ $dx$

• 3.
Evaluate the integral (even powers of sine/cosine).
a)
$\int \sin^2x$ $dx$

b)
$\int \cos^4x$ $dx$

c)
$\int \sin^4x$ $\cos^2 x$ $dx$