\( \int\displaystyle\frac{\cos 5x+\cos 4x}{1-2\cos 3x}\,dx \) is equal to [\( C \) is an arbitrary constant]
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( -\displaystyle\frac{\sin 2x}{2}-\sin x+c \)
The numerator is a multiple of the denominator. Using \( 2\cos A\cos B=\cos(A+B)+\cos(A-B) \),
\( (1-2\cos 3x)(\cos 2x+\cos x) \)
\( =\cos 2x+\cos x-(\cos 5x+\cos x)-(\cos 4x+\cos 2x) \)
\( =-(\cos 5x+\cos 4x) \)
So the integrand equals \( -(\cos 2x+\cos x) \):
\( \displaystyle \int-(\cos 2x+\cos x)\,dx=-\frac{\sin 2x}{2}-\sin x+c \)
NCERT Exemplar:
Integrals
