please solve this without using reduction formula - Maths - Indefinite and Definite Integration

Question

please solve this without using reduction formula

Atul Shaw · Accepted Answer

Dear student,∫cos7 x dx=∫cos6 x×cosx dx=∫(cos2 x)3×cos x dx   =∫(1-sin2 x)3×cos x dx   [cos2x=1-sin2 x]Let sin x=u   ⇒cosx dx=du=∫(1-u2)3 du=∫(1-u6+3u4-3u2) du    [(a-b)3=a3-b3+3ab2-3a2b]=∫du-∫u6 du+∫3u4 du-∫3u2 du=u-u77+3u55-3u33+c=-sin7 x7+3 sin5 x5-sin3 x+sin x+cRegards