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.

(See Period of Sine and Cosine if you are not sure about this.)

Effective Current

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.

This is a graph of our cosine current with the RMS effective current shown.

Graph of i(t)=3+2cos(t), with the RMS current indicated by the dashed magenta line

Graph of `i(t)=3+2cos(t)`, with the RMS current indicated by the dashed magenta line.

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`

The solution for Exercise 1 represents the area under the curve `y(x)=sqrt(cos\ x)\ sin^3x\ dx` between `pi/3 <= x <= pi/2`. Here is that graph:

Graph of y(x)=sqrt(cos x) sin^3x dx, indicating the area under the curve between pi/3 <= x <= pi/2

Graph of `y(x)=sqrt(cos x)\ sin^3x`, indicating the area under the curve from `x=pi/3` to `x=pi/2`.

Zooming out that graph shows it's periodic (it repeats itself) with period `2pi`. There are holes in the graph because of the `sqrt(cos x)` part (we can't have the square root of a negative number).

Graph of y(x)=2/(1+sin(x)), indicating the area under the curve from x=pi/6 to x=pi/3

Graph of `y(x)=sqrt(cos x)\ sin^3x dx`, zoomed out to see its periodic nature.

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

Here is the graph of the integration we just found, indicating the area under the curve `y=sin^2 4x`.

Graph of y(x)=sin^2 4x, indicating the area under the curve from x=0 to x=1

Graph of `y(x)=sin^2 4x`, indicating the area under the curve from `x=0` to `x=1`.

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

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

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

Graph of y(x)=2/(1+sin(x)), indicating the area under the curve from x=pi/6 to x=pi/3

Graph of `y(x)=2/(1+sin(x))`, indicating the area under the curve from `x=pi/6` to `x=pi/3`.

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`.

Here's the graph of the arc length we just found. I needed to take the absolute value of the `cos(x)` values, otherwise there would be gaps in the graph (whenever `cos(x)` was negative).

Graph of second roder differential equation using software

Graph of `y(x)=ln|cos x|`, with the curve length we just found indicated in magenta.