Properties of Integration

In summary, the conversation discussed the relationship between trigonometric functions and their complementary functions, and the proof of this relationship. It also mentioned limitations to this property, and requested a formal proof for the identities involving sine and cosine. A readily accessible proof for these identities was provided, as well as a link to the basic identities for the sine and cosine of the sum and difference of two angles.
  • #1
MathewsMD
433
7
Hi,

Recently in class, my professor went over a relationship that exists between trigonometric functions, T(x), and their complementary functions. That is:

∫ T(x)dx = W(x) + C
∫ coT(x)dx = -co[W(x)] + C

Without really providing a proof, we were told this relationship. I've plugged in numbers and tried graphing the scenario and it works from what I've done so far. If anyone could provide me a formal proof through any source, that would be great since I need a bit more convincing.

Also, is a limitation to this property that both W(x) and T(x) must be a combination trigonometric functions themselves? Since when trying to solve ∫cscxdx using the answer from ∫secdx, an answer cannot be found. Are there any other limitations?

Also, regarding the identities:
sinAcosB = 1/2 [sin(A-B) + sin(A+B)
sinAsinB = 1/2 [cos(A-B) - cos(A+B)]
cosAcosB = 1/2[cos(A-B) + cos(A+B)]

Is there a formal proof readily accessible for these identities?

Thanks!
 
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  • #2
Comes from the definitions of the trigonometric functions on the unit circle... which tells you how they are related to each other.

For the identities:
Helps to noticing that for a point (x,y) in the complex plane, the vector from the origin to (x,y) makes an angle ##\theta## to the real axis:

##e^{i\theta}=\cos\theta+i\sin\theta = x+iy##

You can use that for the integral of trig things too, since ##\int e^x dx = e^x +c##
 
  • #3
MathewsMD said:
Also, regarding the identities:
sinAcosB = 1/2 [sin(A-B) + sin(A+B)
sinAsinB = 1/2 [cos(A-B) - cos(A+B)]
cosAcosB = 1/2[cos(A-B) + cos(A+B)]

Is there a formal proof readily accessible for these identities?

Thanks!

Yes, all it takes is a little manipulation of the basic identities for the sine and cosine of the sum and difference of two angles:

http://en.wikipedia.org/wiki/List_of_trigonometric_identities
 
  • #4

1. What is the definition of integration?

Integration is a mathematical operation that calculates the area under a curve in a given interval. It is the reverse process of differentiation and is used to find the original function from its derivative.

2. What are the properties of integration?

There are several properties of integration, including linearity, the constant multiple rule, the power rule, and the substitution rule. These properties allow us to simplify and solve complex integration problems.

3. How do I use integration to find the area under a curve?

To find the area under a curve, you can use the definite integral, which involves setting the limits of integration and plugging them into the integrand. The resulting value is the area under the curve between the given limits.

4. Can integration be used for more than just finding areas?

Yes, integration has many applications in physics, engineering, and other fields. It can be used to solve problems involving velocity, acceleration, volume, and more.

5. Is there a specific method for solving integration problems?

There are various methods for solving integration problems, including substitution, integration by parts, and trigonometric substitution. The best method to use depends on the complexity of the problem and the type of function being integrated.

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