Reduction formulas for integral of sin and cos

In summary, the reduction formula for n=6 is used to simplify the integration of sin2(x)cos4(x)dx. By using this formula, the integral can be rewritten as cos4(x)-cos6(x)dx. When n=6, the reduction formula becomes (1/6)cos5(x)sinx+(5/6)integral cos4(x)dx. However, in the specific case of sin2(x)cos4(x)dx, the integral is rewritten as (-1/6)cos5(x)sinx+(1/6)integral cos4(x)dx, causing confusion as to why the coefficients change.
  • #1
livvy07
6
0
Integrate sin2(x)cos4(x)dx using reduction formulas?
My book says integral sin2(x)cos4(x)dx= integral cos4(x)-integral cos6x dx
Now the reduction formula for n=6
for integral cos6(x)dx= (1/6)cos5(x)sinx+(5/6) integral cos4(x)dx

Here is the part I don't get: It then says :
sin2(x)cos4(x)dx
=(-1/6)cos5(x)sinx+(1/6) integral cos^4(x)dx

I don't get how the 5/6 becomes 1/6 or why 1/6 becomes -1/6 if that makes any sense? Any help would be great! I've been looking at it for awhile, but I am not seeing it for some reason.
 
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  • #2
hi livvy07! :smile:

(have an integral: ∫ :wink:)

i don't understand that :confused:

∫ sin2(x)cos4(x) dx

= ∫ [sin(x)][sin(x)cos4(x)] dx

then integrate by parts (integrating the second bracket, and differentiating the first bracket)​
 

Related to Reduction formulas for integral of sin and cos

What is a reduction formula for the integral of sin and cos?

A reduction formula is a mathematical tool used to reduce the complexity of an integral. In the case of sin and cos, it is used to simplify integrals involving powers of these functions.

How do you derive a reduction formula for the integral of sin and cos?

To derive a reduction formula, we use integration by parts and algebraic manipulations to reduce the power of the trigonometric functions in the integral. This process is repeated until the integral becomes solvable in terms of known trigonometric functions.

Why are reduction formulas useful in calculus?

Reduction formulas allow us to solve integrals that would otherwise be difficult or impossible to solve. They also provide a general solution for integrals involving powers of sin and cos, making it easier to solve similar integrals in the future.

Can reduction formulas be applied to other trigonometric functions?

Yes, reduction formulas can be applied to other trigonometric functions such as tan, sec, csc, and cot. Similar to sin and cos, these formulas are derived using integration by parts and algebraic manipulations.

Are there any limitations to using reduction formulas for integrals of sin and cos?

Reduction formulas are limited to integrals involving powers of sin and cos. They cannot be applied to integrals involving other functions or more complex trigonometric expressions. Additionally, the reduction process can become tedious and time-consuming for higher powers.

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