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.

Didn't find what you are looking for on this page? Try search:

Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Ready for a break?

Play a math game.

(Well, not really a math game, but each game was made using math...)

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.

Calculus Lessons on DVD

Easy to understand calculus lessons on DVD. See samples before you commit.