Tìm nguyên hàm của hàm số sau:∫cosxcos5xcos7xdx

Question

\(\int \cos x\cos 5x\cos 7x{\rm d}x\)

Answer 1

Ta có:

\(\int \cos x.\cos 5x.\cos 7x{\rm d}x =\frac{1}{2} \int \left(\cos 12x+\cos 2x\right)\cos x{\rm d}x=\frac{1}{2} \int \left(\cos 12x.\cos x+\cos 2x.\cos x\right){\rm d}x\)
=\(\frac{1}{4} \int \left(\cos 13x+\cos 11x+\cos 3x+\cos x\right){\rm d}x= \frac{1}{4} \left(\frac{1}{13} \sin 13x+\frac{1}{11} \sin 11x+\frac{1}{3} \sin 3x+\sin x\right)+C\)
\(=\frac{1}{52} \sin 13x+\frac{1}{44} \sin 11x+\frac{1}{12} \sin 3x+\frac{1}{4} \sin x+C\)