Stuck on this integral (cos and sin)

In summary, when integrating (sin5x)(cos4x)dx, you can use the substitution u=cos(x) and convert the integral to a polynomial. However, when considering definite integrals, be sure to adjust the overall sign appropriately. There are also various rules for integrating powers of sine and cosine.
  • #1
jumbogala
423
4

Homework Statement


Integrate (sin5x)(cos4x)dx


Homework Equations





The Attempt at a Solution


I tried substituting u as (sin5x), but wound up with something way more complicated than what I started with. Same thing happens if I substitute cos^4 for u.

I have no idea what to try next... is there some way that I can break up the integral?
 
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  • #2
Yes. Pick u=cos(x). du=(-sin(x))*dx. That leaves you with sin(x)^4. Use cos(x)^2=1-sin(x)^2.
 
  • #3
Then I end up with

-(1-u^2)(1-u^2)(u^4)du

That doesn't seem any easier to integrate.
 
  • #4
I consider polynomials pretty easy to integrate, but maybe that's just me. Multiply it out.
 
  • #5
Ooh I see, when I multiply it out I get just 1 there. So it's the integral of

1 + (u^4)

Which gives

u + (u^5)/5 or cos x + (cos^5(x))/5. Multiplied by a negative, of course.

But the integral is actually definite and when I plug in the bounds it's giving me the wrong answer.
 
  • #6
You don't get 1+u^4. You get a more complicated polynomial. But not all THAT much more complicated. It only has three terms.
 
  • #7
Ok I think I have it.

(cos^5(x))/5 - (2cos^7(x))/7 + (cos^9(x))/(9)

just wondering: did you know to use u = cos x just by doing lots of practice and recognizing a pattern, or is there a rule you used?
 
  • #8
Yeah, basically. Watch your overall sign there.
 
  • #9
jumbogala said:
Ok I think I have it.

(cos^5(x))/5 - (2cos^7(x))/7 + (cos^9(x))/(9)

just wondering: did you know to use u = cos x just by doing lots of practice and recognizing a pattern, or is there a rule you used?

It's a standard thing. If you've got an odd power of cos or sin times an even power of the other one, you peel off one of the odd power to save for the du and then just convert it to a polynomial. As you did. And check the sign again.
 
  • #10
Oh yeah, forgot to multiply by the negative. The sign of each term should just be flipped, then.

Thanks for your help, I really appreciate it.
 
  • #11
What if sin and cos are both even powers? Or both odd, for that matter?
 

1. What is the best way to approach a stuck integral involving cosine and sine?

The best way to approach a stuck integral involving cosine and sine is to first try using trigonometric identities to simplify the integral. If that does not work, you can try using integration by parts or substitution. It is also helpful to check if the integral can be rewritten in terms of other trigonometric functions.

2. How can I determine the limits of integration for a stuck integral with cosine and sine?

The limits of integration can typically be determined by looking at the original function or problem that the integral represents. If the problem involves a physical situation, the limits may represent the boundaries of the system. If the function is given in terms of x, the limits will be the values of x that correspond to the start and end points of the desired interval.

3. What is the best approach for evaluating a stuck integral with both cosine and sine terms?

The best approach for evaluating a stuck integral with both cosine and sine terms is to use integration by parts. This method involves choosing one of the trigonometric functions as the "u" term and the other as the "dv" term, and then applying the integration by parts formula. This will allow you to reduce the integral to a simpler form that can be evaluated using basic integration techniques.

4. How can I check if my solution to a stuck integral with cosine and sine is correct?

You can check the solution to a stuck integral with cosine and sine by taking the derivative of your solution and seeing if it matches the original integrand. Another way to check is by using a graphing calculator or software to graph both the original function and your solution, and see if they match up.

5. Can I use a calculator to evaluate a stuck integral involving cosine and sine?

Yes, you can use a calculator to evaluate a stuck integral involving cosine and sine. However, it is important to note that calculators may not always give the most simplified form of the integral, so it is recommended to also check the solution by hand. Additionally, some calculators may have limitations on the types of trigonometric functions they can handle, so it is important to check the user manual or online resources for specific instructions on using your calculator for integrals involving cosine and sine.

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