Integration problems. (Integration by parts)

In summary, the conversation is about solving the question \intcos^2(x)dx and the different methods that can be used to solve it. The suggested method is to use the identity sin^2(x) = 1 - cos^2(x) to simplify the integral and then solve using integration by parts. Another method suggested is to use the identity cos^2(x) = (1 + cos(2x))/2. The conversation ends with the clarification that the formula 1/nsinxcos^n-1(x) +n-1/n \intsin^n-2(x)dx can be used for both solving and integrating cos^2(x)dx.
  • #1
BioCore

Homework Statement



Hi, I have a test coming up soon so I was doing some questions from the textbook when I stumbled upon this one and I'm stuck after like 5 tries. Here is the question:

[tex]\int[/tex]cos^2(x)dx

Solve.

Homework Equations



the question then states we should solve using this:
cos^2(x)dx = (cos x)(cos x)

which gives us:
[tex]\int[/tex]cos^2(x)dx = sinxcosx + [tex]\int[/tex]sin^2(x)dx

finally we should use sin^2(x) + cos^2(x) = 1 to replace the sin^2(x) at the right side of integral.

The Attempt at a Solution


so basically I tried using integral by parts, since we are studying this topic currently
and set:

u= 1 - cos^2(x) and dv = dx
du = 2cosxsinx and v = x

When I plug in the values into the integration and try to solve I don't end up with their answer. I must be overlooking something and this is where I am stuck:

[tex]\int[/tex]cos^2(x)dx = sinxcosx + x(1-cos^2x) - [tex]\int[/tex]2xcosxsinx

The final answer should be:
[tex]\int[/tex]cos^2(x)dx = 1/2sinxcosx + 1/2x + C

Thanks for the help.
 
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  • #2
All problems of that type can be solved using this "formulas" :

[tex]\int[/tex]cos^n(x)dx = 1/nsinxcos^n-1(x) +n-1/n [tex]\int[/tex]sin^n-2(x)dx

[tex]\int[/tex]sin^n(x)dx = -1/nsin^n-1(x)cosx +n-1/n [tex]\int[/tex]sin^n-2(x)dx
 
  • #3
well, you could also use this identity

cos^2(x)=(1+cos(2x))/2, it would really help you get to the answer pretty quickly, and without needing to do ineg by parts at all.
 
  • #4
However, the point of the method that was suggested originally is that, after the first integration you have, as you say,
[tex]\int cos^2(x)dx = sinxcosx + \int sin^2(x)dx[/tex]
Now let [itex]sin^2(x)= 1- cos^2(x)[/itex] and that becomes
[tex]\int cos^2(x)dx= sin(x) cos(x)+ \int (1- cos^2(x))dx[/tex]
[tex]\int cos^2(x)dx= sin(x) cos(x)+ \int dx- \int cos^2(x) dx[/tex]
[tex]\int cos^2(x)dx= sin(x) cos(x)+ x - \int cos^2(x) dx[/tex]
Add [itex]\int cos^2(x)dx[/itex] to both sides of the equation and you are almost done.
 
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  • #5
Halls off Ivy thanks a lot, I forgot that rule of integration. thanks a lot, yeah now that I tried it out it actually works. Thanks a lot.
 
  • #6
[tex] cos^2(x)dx= 1+cos2x/2[/tex]




wat i want to ask is...is this the formula for cos^2x dx or can it be used wid [tex]\int[/tex] too??
 

Related to Integration problems. (Integration by parts)

1. What is integration by parts?

Integration by parts is a method used to find the integral of a product of two functions. It involves using the product rule from calculus to rewrite the integral in a different form that is easier to solve.

2. When should integration by parts be used?

Integration by parts should be used when the integral of a product of two functions cannot be easily solved by substitution or other methods. It is also useful for integrating functions that are not in standard form.

3. How do you set up an integration by parts problem?

To set up an integration by parts problem, you must identify the two functions that make up the product in the integral. Then, choose one of the functions to differentiate and the other to integrate. This will give you two new functions that can be used to rewrite the integral.

4. What is the formula for integration by parts?

The formula for integration by parts is: ∫u*dv = u*v - ∫v*du, where u and v are the two functions identified in the original integral. This formula is derived from the product rule.

5. Are there any tricks or tips for solving integration by parts problems?

One helpful tip for solving integration by parts problems is to choose the function to differentiate based on its simplicity. This will make the integration process easier. Additionally, if the integral becomes more complex after applying the formula, it may be helpful to use integration by parts multiple times.

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