Integration  x2 cos3 x dx  explain in great detail and give full solution

Asked by haroonrashidgkp | 18th Oct, 2018, 04:30: PM

Expert Answer:

integral x squared cos cubed x space d x
u s e space cos space 3 x equals 4 cos cubed x minus 3 cos x rightwards arrow cos cubed x equals fraction numerator cos space 3 x space plus 3 cos x over denominator 4 end fraction
integral x squared left parenthesis fraction numerator cos space 3 x space plus 3 cos x over denominator 4 end fraction right parenthesis d x

1 fourth left square bracket integral x squared cos 3 x space d x space plus integral 3 x squared cos x space d x
u sin g space i n t e g r a t i o n space b y space p a r t s
integral u. v d x space equals u integral v d x minus integral left square bracket fraction numerator d u over denominator d x end fraction integral v d x right square bracket d x
1 fourth left square bracket x squared space fraction numerator sin 3 x over denominator 3 end fraction minus integral left square bracket 2 x space. fraction numerator sin 3 x over denominator 3 end fraction right square bracket d x plus 3 space left parenthesis x squared sin x minus integral 2 x space sin x space d x right parenthesis right square bracket
1 fourth left square bracket x squared space fraction numerator sin 3 x over denominator 3 end fraction minus left parenthesis 2 over 3 x. negative fraction numerator cos 3 x over denominator 3 end fraction minus integral 2 over 3. negative fraction numerator cos 3 x over denominator 3 end fraction d x right parenthesis plus 3 left parenthesis x squared sin x minus left parenthesis 2 x. negative cos x minus integral 2. negative cos x d x right parenthesis right parenthesis right square bracket
1 fourth left square bracket x squared space fraction numerator sin 3 x over denominator 3 end fraction plus 2 over 9 x. cos 3 x minus 2 over 9 fraction numerator sin 3 x over denominator 3 end fraction plus 3 x squared sin x plus 6 x cos x minus 6 sin x right square bracket
1 fourth left square bracket x squared space fraction numerator sin 3 x over denominator 3 end fraction plus 2 over 9 x. cos 3 x minus fraction numerator 2 sin 3 x over denominator 27 end fraction plus 3 x squared sin x plus 6 x cos x minus 6 sin x right square bracket

Answered by Ram Singh | 19th Oct, 2018, 06:18: PM