Study the definitions of the six common Trigonometric Integrals. Being familiar with these will save time when integrating trigonometric functions. If u is a function of x, then:
The integral of (sin u · u' dx ) = the integral of (sin u du ) = 'cos u + C
The integral of (cos u · u' dx) = the integral of (cos u du) = sin u + C
The integral of (sec^2 u · u' dx) = the integral of (sec^2 u du) = tan u + C
The integral of (csc^2 u · u' dx) = the integral of (csc^2 u du) = 'cot u + C
The integral of (sec u tan u · u' dx) = the integral of (sec u tan u du) = sec u + C
The integral of (csc u cot u · u' dx) = the integral of (csc u cot u du) = 'csc u + C
Where: sin = sine, cos = cosine, sec = secant, csc = cosecant, tan = tangent, cot = cotangent
Learn the properties of the Indefinite Integral. Knowing these properties may help when solving an integration of a trigonometric integral, a simple substitution for part of a problem or the entire solution may result. If u(x) and v(x) are integrable, then:
The integral of x^n dx = [x^(n+1) / (n + 1)] + C (n is not equal to -1)
The intergral of cv(x) dx = c times the integral of v (x) dx where c is a constant
The integral of [u(x) + v(x)] dx = the integral of u(x) dx + the integral of v(x) dx
Study the Basic Trigonometric Identities to become familiar with these definitions. Recognizing them is key in solving trigonometric integrals. Listed here are the most common 12. An exhaustive list can be found in a calculus text book. See the Resource Section of this article for one such text.
tan x = sin x / cos x
cot x = cos x / sin x
sec x = 1 / cos x
csc x = 1 / sin x
sin^2 x + cos^2 x = 1
tan^2 x + 1 = sec^2 x
1 + cot^2x = csc^2 x
sin^2 x = 1/2 (1 - cos 2x)
sin 2x = 2 sin x cos x
cos^2 x = (cos 2x + 1) / 2
tan 2x = (2 tan x / 1 - tan^2 x
cos^2 x = 1/2 (1 + cos 2x)
Consider the integral to be evaluated. This is the first step in evaluating or solving any trigonometric integration. Note the form and the trigonometric function of the integral to be evaluated. This step will lead to the approach to use for the solution.
Determine if the integral to be evaluated is in the same form as any of the derived forms. To do this, compare the given integral to be solved to the derived forms. If it is in the same form as any of the derived integrals, simply substitute. For example, to find the integral of sin x dx, look at the list or recall the definitions, and ask yourself whether this integral has already been derived. Looking at the list of definitions reveals that sin x dx is defined. From the definition, write "the integral of sin x dx = - cos x + C."
Manipulate the given expression if possible to achieve a derived form if it is not given directly. For example, to evaluate the integral of x cos x^2 dx, recall from the list of definitions of trigonometric integrals that the integral of cos u u' dx equals the integral of cos u du, which equals sin u + C. In the example, u equals x^2 and u' equals 2x. Remember that u' is defined as the derivative of u. For a full explanation of the Fundamental Theorem of Calculus and derivatives, see the calculus text noted in the Resource Section of this article. The equation must be manipulated to get rid of the 2 in the 2x because it does not appear in the given definition. To do this, multiply both sides of the integral by 1/2. Doing this does not change the value of the expression but achieves the correct form.
This expression becomes "the integral of x cos x^2 dx = 1/2 times the integral of (1/2) 2x cos x^2 dx" or "the integral of x cos x^2 dx = 1/2 times the integral of x cos x^2 dx."
Solve the resulting integral by using the derived form. The example yields "the integral of x cos x^2 dx = 1/2 times the integral of x cos x^2 dx." From the list of definitions of trigonometric integrals, if u is a function of x, then:
the integral of cos u u' dx = the integral of cos u du
the integral of cos u du = sin u + C.
Using the derived form solve the equation:
the integral of x cos x^2 dx = 1/2 sin x^2 + C.
Evaluate the integral sin^2 x cos^3 x dx. Note that it is of the form sin^m u cos^n u u' dx. Integrals of this form are defined as "the integral of sin^m u cos^n u u' dx = the integral of sin^m u cos^n u du."
Define the "m" and "n" terms. In this example, the exponent in the m position is 2 and the exponent in the n position is 3.
Determine if n is an odd positive integer or if m is an odd positive integer or if m and n are both positive even integers. In this example n = 3 and m = 2. Therefore n is an odd positive integer. If m were an odd positive integer as well, the expression would be rewritten with (m - 1) as the exponent in place of m. For example, if m were 3 then (3 - 1) or 2 would be the new exponent for the m term and the trigonometric identity used would be: sin^2 u = 1 - cos^2 u. If m and n are both positive even integers use the identities "sin^2 x = (1 - cos 2x) / 2 and cos^2 x = ( 1 + cos 2x) / 2" to get odd powers of the cosine.
Rewrite the expression as "sin^m u cos^n u = sin^m u cos^(n -1) u cos u." Specifically, write
"sin^2 x cos^(3 - 1) x = sin^2 x cos^2 x cos x dx."
Use the identity cos^2 x = 1 - sin^2 x to get the expression in the correct form. Using the example:
the integral of sin^2 x cos^3 x dx = the integral of sin^2 x cos^2 x cos x dx
the integral of sin^2 x cos^2 x cos x dx = the integral of sin^2 x (1 - sin^2 x) cos x dx
the integral of sin^2 x (1 - sin^2 x) cos x dx = the integral of (sin^2 x - sin^4 x) cos x dx
the integral of (sin^2 x - sin^4 x) cos x dx = [(sin^3 x ) / 3] - [(sin^5 x) / 5] + C.
Recall the Common Substitutions:
Form: (a^2 -u^2)^(1/2); Substitution: u = a sin x; Identity: cos^2 x = 1 - sin^2 x
Form: (a^2 + u^2)^(1/2); Substitution: u = a tan x; Identity:sec^2 x = 1 + tan^2 x
Form: (u^2 - a^2)^(1/2); Substitution: u = a sec x; Identity: tan^2 x = sec^2 x - 1
Evaluate the given expression. For example, evaluate the integral from 0 to 3 of (9 - x^2)^(1/2) dx. This expression is of the form (a^2 - u^2)^1/2 so the substitution will be u = a sin x and the identity used will be cos^2x = 1 - sin^2 x. In this case, a = 3 because 3^2 = 9.
Make the substitution. Because the interval goes from zero to three begin by making the substitution x = 3 sin @ into the original equation. This yields: dx = 3 cos @ d@. Because the integral goes from 0 to 3 consider when x = 0 , sin @ = 0 and @ = 0 when x = 3, sin @ = 1 and @ = (3.14)/2.
Solve the Integral. For example, the integral from 0 to 3 of (9 - x^2)^(1/2) dx
= the integral from 0 to (3.14)/2 of (9 - 9 sin^2 @)^(1/2) 3 cos @ d@
= the Integral from 0 to (3.14/2) of 9 cos^2 @ d@
= 9/2 times the Integral from 0 to (3.14)/2 of ( 1 + cos 2@) d@.
= 9/2 (@ + (1/2) sin 2@) from 0 to (3.14)/2
= 9/2 [(3.14)/2 + 1/2 sin @) - (0 + 1/2 sin 0)]
= (9 (3.14) )/ 4