Reduction Formula and Integration by Parts

In summary: CIn summary, the conversation is about finding the integral of sin^2x and using trigonometric identities to simplify the process. The final answer is x/2 - 1/4sin(2x) + C.
  • #1
Serena_Greene
9
0
I was sick and missed class:rolleyes:

∫ sin^2x dx = x/2 – sin2x/4 + C
(I see that there is a trig identity in the answer as sin2x = 2sinxcosx)

What I tried (copying the example in the book)
u = sinx du = sinxcosx dx (why is du not cosx?)
v = -cos x dv = sinx dx (why is this not just dx?)

∫ sin^2x dx = -cosxsinx + 1 ∫ sinx cos^2x dx (okay why did I add +1 and where did the funky ∫ come from) I see that cos^2x = 1 – sin^2x.

So now (from the example in the book but why?)
∫ sin^2x dx = -cosxsinx + 1 ∫ sinx dx – 1 ∫ sin^2x dx

2 ∫ sin^2x dx = -cosxsinx + 1 ∫ sinx dx (what happened to the -1?)

∫ sin^2x dx = -½cosxsinx + 1/2 ∫ sinx dx

Which I thought should give

-½cosxsinx – ½cosx + C

Nothing like the above answer……. Can someone help?

-Serena
 
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  • #2
I don't want to read trough the whole thing, but I think it's easier to use the trig identity from the start:

sin^2 x = 1/2 - 1/2 cos(2x)

edit: this way you'll get to the answer easily too :)
 
Last edited:
  • #3
u = sin(x)
du = cos)dx
v = -cos(x)
dv = sin(x)dx

Int[sin^2(x)] = -sin(x)cos(x) + Int[cos^2(x)]
Int[sin^2(x)] = -1/2sin(2x) + Int[1] - Int[sin^2x]
2Int[sin^2(x)] = -1/2sin(2x) + x
Int[sin^2(x)] = x/2 - 1/4sin(2x)
 

Related to Reduction Formula and Integration by Parts

1. What is a reduction formula?

A reduction formula is a mathematical technique used to reduce a complex integral into a simpler form through repeated application of integration by parts. It is particularly useful for integrals involving products of trigonometric, exponential, or logarithmic functions.

2. How is integration by parts related to reduction formula?

Integration by parts is the basis for reduction formula. It involves breaking down a complex integral into two simpler parts and using a specific formula to integrate one part while differentiating the other. By repeating this process, a reduction formula can be derived.

3. What are the steps for using reduction formula?

The steps for using reduction formula are as follows:

  1. Identify the integral to be solved.
  2. Use integration by parts to split the integral into two parts.
  3. Apply the reduction formula to simplify the integral.
  4. Repeat the process until the integral can be easily solved.
  5. Use the final result to solve the original integral.

4. When should I use reduction formula?

Reduction formula is most useful when dealing with integrals that involve products of trigonometric, exponential, or logarithmic functions. It can also be used to simplify integrals with powers of polynomials or inverse trigonometric functions.

5. What are some common mistakes to avoid when using reduction formula?

Some common mistakes to avoid when using reduction formula include:

  • Forgetting to apply the reduction formula more than once.
  • Not using the correct formula for the given integral.
  • Incorrectly differentiating or integrating the parts of the integral.
  • Not simplifying the integral after applying the reduction formula.
  • Forgetting to use the final result to solve the original integral.

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