integral subscript 0 superscript straight pi divided by 2 end superscript left parenthesis cos 5 x. sin 3 x right parenthesis d x divided by cos x space space=?

Asked by Sunil Soni | 7th May, 2016, 11:44: PM

Expert Answer:

C o n s i d e r space t h e space g i v e n space i n t e g r a l
I equals integral subscript 0 superscript straight pi over 2 end superscript fraction numerator open parentheses cos 5 x times sin 3 x close parentheses over denominator cos x end fraction d x


W e space k n o w space t h a t comma space
sin 3 x equals 3 sin x minus 4 sin cubed x
cos 5 x equals cos to the power of 5 x minus 10 cos cubed x sin squared x plus 5 cos x sin to the power of 4 x
T h u s comma
c o s 5 x times s i n 3 x equals open parentheses c o s to the power of 5 x minus 10 c o s cubed x s i n squared x plus 5 c o s x s i n to the power of 4 x close parentheses open parentheses 3 s i n x minus 4 s i n cubed x close parentheses
space space space space space space space space space space space space space space space space space space space space space space equals sin x open parentheses c o s to the power of 5 x minus 10 c o s cubed x open parentheses 1 minus cos squared x close parentheses plus 5 c o s x open parentheses 1 minus c o s squared x close parentheses squared close parentheses open parentheses 3 minus 4 open parentheses 1 minus c o s squared x close parentheses close parentheses
space space space space space space space space space space space space space space space space space space space space space space equals s i n x open parentheses c o s to the power of 5 x minus 10 c o s cubed x open parentheses 1 minus c o s squared x close parentheses plus 5 c o s x open parentheses 1 plus c o s to the power of 4 x minus 2 cos squared x close parentheses close parentheses open parentheses 3 minus 4 plus 4 cos squared x close parentheses
space space space space space space space space space space space space space space space space space space space
T h u s comma
fraction numerator open parentheses c o s 5 x times s i n 3 x close parentheses over denominator c o s x end fraction equals fraction numerator s i n x open parentheses c o s to the power of 5 x minus 10 c o s cubed x open parentheses 1 minus c o s squared x close parentheses plus 5 c o s x open parentheses 1 plus c o s to the power of 4 x minus 2 c o s squared x close parentheses close parentheses open parentheses 3 minus 4 plus 4 c o s squared x close parentheses over denominator cos x end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space equals s i n x open parentheses c o s to the power of 4 x minus 10 c o s squared x open parentheses 1 minus c o s squared x close parentheses plus 5 open parentheses 1 plus c o s to the power of 4 x minus 2 c o s squared x close parentheses close parentheses open parentheses negative 1 plus 4 c o s squared x close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space equals s i n x open parentheses c o s to the power of 4 x minus 10 c o s squared x plus 10 c o s to the power of 4 x plus 5 plus 5 c o s to the power of 4 x minus 10 c o s squared x close parentheses open parentheses negative 1 plus 4 c o s squared x close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space equals s i n x open parentheses 16 c o s to the power of 4 x minus 20 c o s squared x plus 5 close parentheses open parentheses negative 1 plus 4 c o s squared x close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space equals s i n x open parentheses 64 c o s to the power of 6 x minus 96 c o s to the power of 4 x plus 40 c o s squared x minus 5 close parentheses

I equals integral subscript 0 superscript straight pi over 2 end superscript s i n x open parentheses 64 c o s to the power of 6 x minus 96 c o s to the power of 4 x plus 40 c o s squared x minus 5 close parentheses d x

L e t space t equals cos x semicolon space d t equals negative sin x d x
W h e n space x equals 0 comma space t equals 1
W h e n space x equals straight pi over 2 comma space t equals 0
T h u s
I equals negative integral subscript 1 superscript 0 open parentheses 64 t to the power of 6 minus 96 t to the power of 4 plus 40 t squared minus 5 close parentheses d t
space space space equals negative open square brackets negative integral subscript 0 superscript 1 open parentheses 64 t to the power of 6 minus 96 t to the power of 4 plus 40 t squared minus 5 close parentheses d t close square brackets
space space space space equals integral subscript 0 superscript 1 open parentheses 64 t to the power of 6 minus 96 t to the power of 4 plus 40 t squared minus 5 close parentheses d t
space space space equals open square brackets 64 cross times t to the power of 7 over 7 minus 96 cross times t to the power of 5 over 5 plus 40 cross times t cubed over 3 minus 5 t close square brackets subscript 0 superscript 1
space space space equals open square brackets 64 over 7 minus 96 over 5 plus 40 over 3 minus 5 close square brackets
space space space equals open square brackets fraction numerator 960 minus 2016 plus 1400 minus 525 over denominator 105 end fraction close square brackets
space space equals fraction numerator negative 181 over denominator 105 end fraction
equals negative 1.723809523

Answered by Vimala | 8th May, 2016, 09:00: PM