Trigonometric Integrals

INTEGRATION OF TRIGONOMETRIC INTEGRALS

Recall the definitions of the trigonometric functions.

$\displaystyle{ \tan x = { \sin x \over \cos x } }$
$\displaystyle{ \sec x = { 1 \over \cos x } }$
$\displaystyle{ \cot x = { \cos x \over \sin x } = { 1 \over \tan x } }$
$\displaystyle{ \csc x = { 1 \over \sin x } }$

The following indefinite integrals involve all of these well-known trigonometric functions. Some of the following trigonometry identities may be needed.

A.) $\cos^2 x + \sin^2 x = 1$
B.) $\sin 2x = 2 \sin x \cos x$
C.) $\cos 2x = 2 \cos^2 x - 1$ so that $\cos^2 x = \displaystyle{ 1+\cos 2x \over 2}$
D.) $\cos 2x = 1 - 2 \sin^2 x$ so that $\sin^2 x = \displaystyle{ 1-\cos 2x \over 2}$
E.) $\cos 2x = \cos^2 x - \sin^2 x$
F.) $1 + \tan^2 x = \sec^2 x$ so that $\tan^2 x = \sec^2 x - 1$
G.) $1 + \cot^2 x = \csc^2 x$ so that $\cot^2 x = \csc^2 x - 1$

It is assumed that you are familiar with the following rules of differentiation.

$D (\sin x) = \cos x$
$D (\cos x) = - \sin x$
$D (\tan x) = \sec^2 x$
$D (\cot x) = - \csc^2 x$
$D (\sec x) = \sec x \tan x$
$D (\csc x) = - \csc x \cot x$

These lead directly to the following indefinite integrals.

1.) $\displaystyle{ \int \cos x \, \ dx } \ = \ \sin x + C$
2.) $\displaystyle{ \int \sin x \, \ dx } \ = \ - \cos x + C$
3.) $\displaystyle{ \int \sec^2 x \, \ dx } \ = \ \tan x + C$
4.) $\displaystyle{ \int \csc^2 x \, \ dx } \ = \ - \cot x + C$
5.) $\displaystyle{ \int \sec x \tan x \, \ dx } \ = \ \sec x + C$
6.) $\displaystyle{ \int \csc x \cot x \, \ dx } \ = \ - \csc x + C$

The next four indefinite integrals result from trig identities and u-substitution.

7.) $\displaystyle{ \int \tan x \, \ dx } \ = \ \ln \vert \sec x \vert + C$
8.) $\displaystyle{ \int \cot x \, \ dx } \ = \ \ln \vert \sin x \vert + C$
9.) $\displaystyle{ \int \sec x \, \ dx } \ = \ \ln \vert \sec x + \tan x\vert + C$
10.) $\displaystyle{ \int \csc x \, \ dx } \ = \ \ln \vert \csc x - \cot x \vert + C$

We will assume knowledge of the following well-known, basic indefinite integral formulas :

$\displaystyle{ \int x^n \,dx } = \displaystyle{ {x^{n+1} \over n+1 } + C }$ , where is a constant $(n \ne -1 )$
$\displaystyle{ \int { 1 \over x } \,dx } = \displaystyle{ \ln \vert x\vert + C }$
$\displaystyle{ \int e^x \,dx } = \displaystyle{ e^x + C }$
$\displaystyle{ \int k f(x) \,dx } = k \displaystyle{ \int f(x) \,dx }$ , where is a constant
$\displaystyle{ \int ( f(x) \pm g(x) ) \,dx } = \displaystyle{ \int f(x) \,dx } \pm \displaystyle{ \int g(x) \,dx }$

Most of the following problems are average. A few are challenging. Many use the method of u-substitution. Make careful and precise use of the differential notation and and be careful when arithmetically and algebraically simplifying expressions.

PROBLEM 1 : Integrate $\displaystyle{ \int { \sin{3x} } \,dx }$ .
Click HERE to see a detailed solution to problem 1.
PROBLEM 2 : Integrate $\displaystyle{ \int { \tan{5x} } \,dx }$ .
Click HERE to see a detailed solution to problem 2.
PROBLEM 3 : Integrate $\displaystyle{ \int {5 \sec{4x} \tan{4x}} \,dx }$ .
Click HERE to see a detailed solution to problem 3.
PROBLEM 4 : Integrate $\displaystyle{ \int { (\sin{x} + \cos{x})^2 } \,dx }$ .
Click HERE to see a detailed solution to problem 4.
PROBLEM 5 : Integrate $\displaystyle{ \int { 3 \cos^2 5x } \,dx }$ .
Click HERE to see a detailed solution to problem 5.
PROBLEM 6 : Integrate $\displaystyle{ \int { (2 + \tan{x})^2 } \,dx }$ .
Click HERE to see a detailed solution to problem 6.
PROBLEM 7 : Integrate $\displaystyle{ \int { \sin^3 x } \,dx }$ .
Click HERE to see a detailed solution to problem 7.
PROBLEM 8 : Integrate $\displaystyle{ \int {\cos{5x} \over 3 + \sin{5x} } \,dx }$ .
Click HERE to see a detailed solution to problem 8.
PROBLEM 9 : Integrate $\displaystyle{ \int { \cos^2 {x} \over 1 + \sin{x} } \,dx }$ .
Click HERE to see a detailed solution to problem 9.
PROBLEM 10 : Integrate $\displaystyle{ \int { \sin{x} \over 1 + \sin{x} } \,dx }$ .
Click HERE to see a detailed solution to problem 10.
PROBLEM 11 : Integrate $\displaystyle{ \int {(\csc{3x} + \cot{3x})^2 } \,dx }$ .
Click HERE to see a detailed solution to problem 11.
PROBLEM 12 : Integrate $\displaystyle{ \int { ( \sec^2 x ) \,\sqrt{ 5 + \tan x} } \,dx }$ .
Click HERE to see a detailed solution to problem 12.
PROBLEM 13 : Integrate $\displaystyle{ \int \tan^5 x \, dx}$
Click HERE to see a detailed solution to problem 13.
PROBLEM 14 : Integrate $\displaystyle{ \int { (\sin x - \cos x) (\sin x + \cos x)^5 } \,dx }$ .
Click HERE to see a detailed solution to problem 14.
PROBLEM 15 : Integrate $\displaystyle{ \int { ( \cos x ) \, e^{4 + \sin x} } \,dx }$ .
Click HERE to see a detailed solution to problem 15.
PROBLEM 16 : Integrate $\displaystyle{ \int {\sin {3x} \ \sin{(\cos{3x})} } \,dx }$ .
Click HERE to see a detailed solution to problem 16.
PROBLEM 17 : Integrate $\displaystyle{ \int { \cos x \, \ln{(\sin x)} \over \sin x } \,dx }$ .
Click HERE to see a detailed solution to problem 17.
PROBLEM 18 : Integrate $\displaystyle{ \int { (\sec x \tan x ) \sqrt {4 + 3 \sec x} } \,dx }$ .
Click HERE to see a detailed solution to problem 18.
PROBLEM 19 : Integrate $\displaystyle{ \int { e^x \cos{(e^x)} } \,dx }$ .
Click HERE to see a detailed solution to problem 19.

Some of the following problems require the method of integration by parts. That is, $\ \ \displaystyle{ \int u \,dv = uv - \int v \,du }$ .
PROBLEM 20 : Integrate $\displaystyle{ \int { x \sin{3x} } \,dx }$ .
Click HERE to see a detailed solution to problem 20.
PROBLEM 21 : Integrate $\displaystyle{ \int { x^2 \cos x } \,dx }$ .
Click HERE to see a detailed solution to problem 21.
PROBLEM 22 : Integrate $\displaystyle{ \int {\sin x \cos x \ \,e^{\sin x} } \,dx }$ .
Click HERE to see a detailed solution to problem 22.
PROBLEM 23 : Integrate $\displaystyle{ \int { e^x \sin{x} } \,dx }$ .
Click HERE to see a detailed solution to problem 23.
PROBLEM 24 : Integrate $\displaystyle{ \int { \sin{3x} \cos{4x} } \,dx }$ .
Click HERE to see a detailed solution to problem 24.
PROBLEM 25 : Integrate $\displaystyle{ \int { \sec x \ \sqrt{ \sec x + \tan x} } \,dx }$ .
Click HERE to see a detailed solution to problem 25.
PROBLEM 26 : Integrate $\displaystyle{ \int { \sin{2x} - \cos{2x} \over \sin{2x} + \cos{2x} } \,dx }$ .
Click HERE to see a detailed solution to problem 26.
PROBLEM 27 : Integrate $\displaystyle{ \int { \sin{x} + \cos{x} \over e^{-x} + \sin x } \,dx }$ .
Click HERE to see a detailed solution to problem 27.