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.)
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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