What is the integral of #sin(101x)sin^99(x)dx#?

1 Answer
Cem Sentin
Jan 11, 2018

#int sin(101x)*(sinx)^99*dx=1/100sin(100x)*(sinx)^100+C#

Explanation:

#int sin(101x)*(sinx)^99*dx#

=#int sin(100x+x)*(sinx)^99*dx#

=#int (sin(100x)*cosx+cos(100x)*sinx)(sinx)^99*dx#

=#int sin(100x)*(sinx)^99*cosx*dx#+#int cos(100x)*(sinx)^100*dx#

=#1/100sin(100x)(sinx)^100#-#int cos(100x)*(sinx)^100*dx#+#int cos(100x)*(sinx)^100*dx#

=#1/100sin(100x)*(sinx)^100+C#

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