# Thread:

1. ## reduction formula

hey there, first time poster, long time struggler...so i need some help with a problem.

it says to find the reduction formula for the integral (e^x)(sin x)^n in terms of the integral (e^x)(sin x)^n-2.

i know its integration by parts but i have no idea what to do for my substitutions because (sin x)^n on its own is difficult enough so how do I incorporate the e^x?

cheers!

2. Originally Posted by jackm7
hey there, first time poster, long time struggler...so i need some help with a problem.

it says to find the reduction formula for the integral (e^x)(sin x)^n in terms of the integral (e^x)(sin x)^n-2.

i know its integration by parts but i have no idea what to do for my substitutions because (sin x)^n on its own is difficult enough so how do I incorporate the e^x?

cheers!
Integration by parts twice. For the first use

$u = \sin^n x,\; dv = e^x\,dx$

so

$du = n \sin ^{n-1}x \cos x\, dx \;\;v = e^x$

and

$e^x \sin^n x - n \int e^x \sin ^{n-1}x \cos x\,dx$

For the second by parts

$u = \sin ^{n-1}x \cos x,\; dv = e^x\,dx$

and use some identites - see how that goes.
{"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": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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": 0.9163363575935364, "perplexity_flag": "middle"}