Indefinite Integration
Integral Calculus-1
star_batch_jee_advanced_2025
Grade 12
Question:
If $I_{m,n} = \int \cos^m x \sin nx dx$, then $7I_{4,3} - 4I_{3,2} =$$
Constant
$$-\cos^2 x + C$$
$$-\cos^4 x \cos 3x + C$$
$$\cos 7x - \cos 4x + C$$
Step-by-Step Solution
Key Concept: Repeated integration by parts combined with trigonometric identities creates a solvable system of integrals.
Integration by parts is applied to $I_{4,3} = \int \cos^3 x \sin 3x dx$. Using the identity $\sin x \cos 3x = -\sin 2x + \sin 3x \cos x$, the integral is reduced to a combination of $I_{4,3}$, $I_{3,2}$, and lower-order terms. Solving the resulting relation $\frac{7}{3}I_{4,3} - \frac{4}{3}I_{3,2} = \frac{\cos 3x \cos^3 x}{3} + C$ yields the final answer.
Correct Answer: 3