# Thread:

1. ## Something to do with Reduction Formulae...?

Hello everybody,

I have got into a terrible mess with a question.

In my question, I am asked to derive the reduction formulae for integrating (sin x)^m * (cos x)^n, which are on the bottom of this webpage:

Reduction Formulas

I keep nearly getting there, but not quite near enough! I think I must be doing the reduction part wrong.

Any help would be greatly appreciated!!

Jessica.

2. Originally Posted by j_clough
Hello everybody,

I have got into a terrible mess with a question.

In my question, I am asked to derive the reduction formulae for integrating (sin x)^m * (cos x)^n, which are on the bottom of this webpage:

Reduction Formulas

I keep nearly getting there, but not quite near enough! I think I must be doing the reduction part wrong.

Any help would be greatly appreciated!!

Jessica.
Hello,

What exactly did you do ?

integrate by parts with u=cos^(n-1) (x) and v'=cos(x)*sin^m (x)

this gives :
$I_{m,n}=\frac{\cos^{n-1}(x) \sin^{m+1}(x)}{m+1}+\frac{n-1}{m+1} \int \left(\sin(x) \cos^{n-2}(x)\right)*\left(\sin^{m+1}(x)\right) ~ dx$

$I_{m,n}=\frac{\cos^{n-1}(x) \sin^{m+1}(x)}{m+1}+\frac{n-1}{m+1} \int \cos^{n-2}(x) \sin^m (x) \sin^2(x) ~ dx$

Use the identity $\sin^2(x)=1-\cos^2(x)$ to find $I_{m,n-2}$ and $I_{m,n}$ in the right hand side of the equation =)
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9602250456809998, "perplexity_flag": "middle"}