UPSC NDA I 2024 Mathematics Paper

Concept:Rewrite the integrand by expressing the numerator as a linear combination of the denominator and its derivative to separate into simpler integrals.Explanation:Let the integral be $I = \int \frac{3\cos x + 4\sin x}{2\cos x + 5\sin x}\,dx$.Set $u = 2\cos x + 5\sin x$. Then $du = (-2\sin x + 5\cos x)\,dx$.We find constants $A$ and $B$ such that:$3\cos x + 4\sin x = A(5\cos x - 2\sin x) + B(2\cos x + 5\sin x)$.Compare coefficients:For $\cos x$: $3 = 5A + 2B$.For $\sin x$: $4 = -2A + 5B$.Solve the system. Multiply the first equation by 5: $15 = 25A + 10B$. Multiply the second by 2: $8 = -4A + 10B$.Subtract the second from the first: $7 = 29A$, so $A = \frac{7}{29}$.Substitute $A$ into $3 = 5A + 2B$: $3 = \frac{35}{29} + 2B$. Then $2B = 3 - \frac{35}{29} = \frac{87-35}{29} = \frac{52}{29}$, giving $B = \frac{26}{29}$.Thus the integral becomes:$I = \frac{7}{29}\int \frac{5\cos x - 2\sin x}{2\cos x + 5\sin x}\,dx + \frac{26}{29}\int \frac{2\cos x + 5\sin x}{2\cos x + 5\sin x}\,dx$.The first integral has numerator $du$, so $\int \frac{du}{u} = \ln|u|$. The second integral is $\int dx = x$.Hence $I = \frac{7}{29}\ln|2\cos x + 5\sin x| + \frac{26}{29}x + C$.Comparing with the form $\alpha x + \beta\ln|2\cos x + 5\sin x| + C$, we get $\alpha = 26$.Answer:$\alpha = 26$