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More Lessons for trigonometric identities

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In this lesson, we will look into some techniques of integrating powers of sine, cosine, tangent and secant. We will use trigonometric identities to integrate certain combinations of trigonometric functions.

To integrate an odd power of sine or cosine, we separate a single factor and convert the remaining even power.

If the **power of cosine is odd** (*n* = 2*k* + 1), save one cosine factor and use the identity sin^{2} *x* + cos^{2} *x* = 1 to express the remaining factors in terms of sine:

Let *u* = sin *x* then *du* = cos *x **dx*

If the **power of sine is odd** (*n* = 2*k* + 1), save one sine factor and use the identity sin^{2} *x* + cos^{2} *x* = 1 to express the remaining factors in terms of cosine:

Let *u* = cos *x* then *du* = – sin *x** dx*

Note: If the powers of both sine and cosine are odd, either of the above methods can be used.

** Example: **

Evaluate

** Solution: **

** Step 1: **

Separate one cosine factor and convert the remaining cos^{2} *x* factor to an expression involving sine using the identity sin^{2} *x* + cos^{2} *x* = 1

** Step 2: **

Let *u* = sin *x* then *du* = cos *x **dx*

** Example: **

Evaluate

** Solution: **

** Step 1: **

Separate one sine factor and convert the remaining sin ^{4} *x* factor to an expression involving cos using the identity sin^{2} *x* + cos^{2} *x* = 1

**Step 2**:

Let *u* = cos *x* then *du* = – sin *x** dx*

If the powers of both the sine and cosine are even, use the half-angle identities

** Example: **

Find

** Solution: **

If we write sin^{2} *x* as 1 – cos^{2} *x*, the integral is no simpler to evaluate.

Instead, we use the half-angle formula for

** Example:**

Find

** Solution:**

We write sin^{4} *x* as (sin^{2} *x*)^{2} and use a half-angle formula:

In order to evaluate cos^{2} 2*x*, we use the half angle formula

Trigonometric Integrals - Part 1 of 6

The 'cookie cutter' case of products of odds powers of sine and/or odd powers of cosine is discussed.

The 'cookie cutter' case of products of even powers of sine and even powers of cosine is discussed

The 'cookie cutter' case of products of even powers of secant and powers of tangent is discussed

The 'cookie cutter' case of products of odd powers of tangent and powers of secant is discussed.

Trigonometric Integrals - Part 5 of 6

The 'cookie cutter' case of products of sin(mx) and cos(nx) are shown

3 trigonometric integrals that do not fit any one technique are discussed.

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