Integration of trigonometric functions

In summary, the integration of trigonometric functions involves finding the antiderivatives of functions such as sine, cosine, tangent, and their inverses. Common techniques include using identities to simplify the integrals, substitution methods, and recognizing patterns in standard integrals. Key results include the integrals of sin(x), cos(x), and sec^2(x), which yield -cos(x), sin(x), and tan(x) + C, respectively. Mastery of these concepts is essential for solving various problems in calculus and applied mathematics.
  • #1
Indir
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Integration problem
Was solving a problem in mathematics and came across the following integration. Unable to move further. Can somebody provide answer for the following ( a and b are constants ).
Integ.gif
 

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  • #2
Why don not you try substitution
[tex]a-b \cos x = u[/tex]?
 
  • #3
A good plan to tackle such questions is: remove what disturbs the most! That often helps to get into the problem. If you have trig functions then it is always good to keep the Weierstraß substitution in mind; not here but in general.
 
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  • #4
On its own, just as a trick, ##sinxcosx=\frac{sin2x}{2}##, with simple integral ##\frac{-Cos2x}{2}##
But, yes, that denominator kills it. Maybe Fresh can write an insight on integrating expressions a/b from the respective integrals of a,b , right, Fresh? ;)
 
  • #5
WWGD said:
On its own, just as a trick, ##sinxcosx=\frac{sin2x}{2}##, with simple integral ##\frac{-Cos2x}{2}##
But, yes, that denominator kills it. Maybe Fresh can write an insight on integrating expressions a/b from the respective integrals of a,b , right, Fresh? ;)
The difficulty with integrating products (and likewise quotients) arises from the fact that differentiation is a derivation. The Jacobi identity / Leibniz rule / product rule rules this world and not the chain rule.
$$
D(f\cdot g) = Df \cdot g + f\cdot Dg
$$
We can sometimes use the fact the ##D\sin= \cos## and ##D\cos= -\sin## and in the case of trigonometric functions. Here is an example:
https://www.physicsforums.com/insig...tion/#Integration-by-Parts-–-The-Leibniz-Rule
 
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FAQ: Integration of trigonometric functions

What is the integral of sin(x)?

The integral of sin(x) with respect to x is -cos(x) + C, where C is the constant of integration.

What is the integral of cos(x)?

The integral of cos(x) with respect to x is sin(x) + C, where C is the constant of integration.

How do you integrate sec^2(x)?

The integral of sec^2(x) with respect to x is tan(x) + C, where C is the constant of integration.

What is the integral of tan(x)?

The integral of tan(x) with respect to x is -ln|cos(x)| + C, or equivalently, ln|sec(x)| + C, where C is the constant of integration.

How do you integrate the product of trigonometric functions, like sin(x)cos(x)?

To integrate the product sin(x)cos(x), you can use the identity sin(2x) = 2sin(x)cos(x). Thus, the integral becomes (1/2)∫sin(2x)dx, which evaluates to -1/4cos(2x) + C, where C is the constant of integration.

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