4. Integration: Basic Trigonometric Forms

by M. Bourne

We obtain the following integral formulas by reversing the formulas for differentiation of trigonometric functions that we met earlier:

`int sin\ u\ du=-cos\ u+K`

`int cos\ u\ du=sin\ u+K`

`int sec^2u\ du=tan\ u+K`

`int csc^2u\ du=-cot\ u+K`

We now apply the power formula to integrate some examples.

NOTE: All angles in this section are in radians. The formulas don't work in degrees.

Example 1: Integrate: `inte^xcsc^2(e^x)dx`

Example 2: Integrate: `int(sin(1/x))/(x^2)dx`

Integral of sec x, csc x

These are obtained by simply reversing the differentiation process.

`{:(int sec\ u\ tan\ u\ du=sec\ u+K),(int csc\ u\ cot\ u\ du=-csc\ u+K):}`

Example 3: Integrate: `int csc\ 2x\ cot\ 2x\ dx`

Integral of tan x, cot x

Now, if we want to find `int tan x\ dx`, we note that

`int tan x\ dx=int(sin x)/(cos x)dx`

Let `u=cos x`, then `du=-sin x\ dx`. Our integral becomes:

`{: (int tan x\ dx,=int(sin x)/(cos x)dx),(,=-int(du)/u),(,=-ln |u|+K),(,=-ln |cos x|+K) :}`

Similarly, it can be shown that

`intcot\ x\ dx=ln\ |sin\ x|+K`

Summary of Integrals of Trigonometric Functions

We summarise the tirgonometric integrals as follows:

`{: (inttan\ u\ du,=-ln\ |cos\ u|+K),(intcot\ u\ du,=ln\ |\sin\ u|+K),(intsec\ u\ du,=ln\ |\sec\ u+tan\ u|+K),(intcsc\ u\ du,=ln\ |csc\ u-cot\ u|+K) :}`

Example 4: Integrate: `intx^2cot\ x^3dx`

Example 5: Integrate: `6int_0^1 tan{:x/2:}dx`

Example 6: Find the area under the curve of `y = sin\ x` from `x = 0` to `x=(3pi)/2`.

Exercises

Integrate each of the given functions:

1. `int(sin\ 2x)/(cos^2x)dx`

2. `int_(pi//4)^(pi//3)(1+sec\ x)^2dx`

3. If the current in a certain electric circuit is i = 110 cos 377t, find the expression for the voltage across a 500-μF capacitor as a function of time. The initial voltage is zero. Show that the voltage across the capacitor is 90° out of phase with the current.

We need the following result from electronics, which gives the voltage across a capacitor, where C is the capacitance:

`V_C=1/Cinti\ dt`

4. A force is given as a function of the distance from the origin as

`F=(2+tan\ x)/(cos\ x`

Express the work done by this force as a function of x if W = 0 for x = 0.

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