Integrals of Trigonometric Functions
The integrals of trigonometric functions can be extremely difficult to find. At this point, you should already be convinced that integration is much harder than differentiation. Any combination of products and quotients of trigonometric functions can be differentiated by studiously following the Product Rule and the Quotient Rule. As long as you try, you will find an answer. The same does not stand for integration. Only an extremely limited portion of trigonometric functions can be integrated.
The ones that can be integrated (using methods I'm familiar with) fall into four categories:
1. The integrand (the shit inside the integral sign) can be written as h'(x)F'(h(x)). Such integrals can be tackled using the Substitution Rule, the integral equivalent of the chain rule. LOOK RIGHT.
2. Those on which integration by parts works. If you don't know what that is, we got a link to it on the home page for integral calculus.
3. Derivatives of the six simple trig functions. Not much explanation is required here. The indefinite integral of cos(x) is just sin(x)+C. The indefinite integral of sec(x)^2 is tan(x)+C. (C is some constant. )
4. Derivatives of the six inverse trig functions. Shit like 1/1+x^2 can be integrated IF you recognize it as a derivative as arctan(x). Chances are that you have forgotten those exotic differentiation formulas. Don't worry.
Still remember the Chain Rule? Most important rule in differential calculus. It states that the derivative of F(h(x)) is F'(h(x))h'(x). The product of the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
The substitution rule, or integration by substitution, says that the indefinite integral of F'(h(x))h'(x) is F(h(x)).
Pretty straightforward, right? Many textbooks will try to indoctrinate you with loads of bullshit concerning variables like u and v. FORGET IT! When using the Substitution rule, look in the integrand for TWO things. h(x), and h'(x).
PART OF THE INTEGRAND WILL BE THE DERIVATIVE OF ANOTHER PART.
Identify those two, and problem solved. Following is a list of functions that can be integrated using the Substitution rule. See if you can identify h(x) and h'(x).
1. f(x) = sin(x)*(cos(x))^2.
2. f(x) = x*(x^2+1)^3.
3. f(x) = e^x * (5*e^x)^2.
1. h(x) = cos(x), h'(x) = sin(x).
2. h(x) = x^2+1, h'(x) = x.
3. h(x) = 5*e^x, h'(x) = e^x.
Got them all? Good for you. Remember, it comes with practice. So, PRACTICE, PRACTICE, AND