# 5. Integration: Other Trigonometric Forms

by M. Bourne

We can use the trigonometric identities that we learned earlier to simplify the integration process.

The main identities are shown here for reference:

cos^2x+sin^2x=1

1+tan^2x=sec^2x

1+cot^2x=csc^2x

2\ cos^2x=1+cos\ 2x

2\ sin^2x=1-cos\ 2x

The process that we use involves using the trigonometric ratios to simplify the expression, or to get the expression into a form that can be integrated.

## Integrating a Product of Powers of Sine and Cosine - one power odd

To integrate a product of powers of sine and cosine, we use

cos^2x+sin^2x=1

if at least one of the powers is odd.

### Example 1

Integrate: int3\ cos^3x\ dx.

## Integrating a Product of Powers of Sine and Cosine - powers even

We use

2\ cos^2x=1+cos\ 2x

or

2\ sin^2x=1-cos\ 2x

if the power of sin\ x or cos\ x is even.

### Example 2

Integrate: intcos^2\ 2x\ dx.

### Example 3

Integrate: 6intcot^3x\ dx.

## Application - Root Mean Square Value

The root mean square value of the function y with respect to x is given by:

y_("rms")=sqrt(1/T int_0^T y^2dx

where T is the period of y.

A common use of this concept is effective current. This is the value of the direct current that would produce the same quantity of heat energy in the same time as a certain alternating current. It is used in the design of heaters.

### Example 4

Find the root mean square (rms) value of i = 3 + 2 cos t.

### Example 5

For a current i given by i = i0 sin ωt, show that the root-mean-square of the current for one period is (i_0)/sqrt2.

### Exercises

Integrate each of the given functions:

1. int_(pi//3)^(pi//2)sqrt(cos\ x)\ sin^3x\ dx\

2. int_0^1sin^2 4x\ dx

3. intcot\ 4x\ csc^4 4x\ dx

4. intsqrt(tan\ x)\ sec^4x\ dx

5. int_(pi//6)^(pi//3)(2dx)/(1+sin\ x

### Application - Length of a Curve

The length s of the arc of a curve y = f(x) from x = a to x = b is given by:

s=int_a^bsqrt(1+((dy)/dx)^2dx

Find the length of the curve y = ln (cos x) from x=0 to x=pi/3.

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