# 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 trigonometric 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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