Question:
If $\int(4\sin\theta+\cos\theta)\cot\theta\cos\theta d\theta=2\theta+\sin2\theta+\log(\sin\theta)+f(2\theta)+c$ and $f(0)=\frac{1}{2}$, then $f(x)=$
If $\int(4\sin\theta+\cos\theta)\cot\theta\cos\theta d\theta=2\theta+\sin2\theta+\log(\sin\theta)+f(2\theta)+c$ and $f(0)=\frac{1}{2}$, then $f(x)=$
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Split higher powers of trigonometric functions using $\cos^2\theta = 1-\sin^2\theta$ to quickly isolate standard integrable pieces like $\cot\theta$.
Updated On: Jun 3, 2026
- $\frac{1}{2}\cos x$
- $\frac{1}{2}\sin x$
- $\frac{1}{2}\sin x\cos x$
- $\frac{\sin^{2}x}{2}$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
We expand the trigonometric integrand by converting cotangent into cosines and sines: $\cot\theta = \frac{\cos\theta}{\sin\theta}$.
Step 2: Meaning
The integrand is $(4\sin\theta + \cos\theta)\frac{\cos^2\theta}{\sin\theta} = 4\cos^2\theta + \frac{\cos^3\theta}{\sin\theta}$. We rewrite these components using standard identities: 1. $4\cos^2\theta = 2(1+\cos2\theta) = 2 + 2\cos2\theta$. 2. $\frac{\cos^3\theta}{\sin\theta} = \frac{\cos\theta(1-\sin^2\theta)}{\sin\theta} = \frac{\cos\theta}{\sin\theta} - \sin\theta\cos\theta = \cot\theta - \frac{1}{2}\sin2\theta$.
Step 3: Analysis
Now integrate each term with respect to $\theta$: $\int (2 + 2\cos2\theta + \cot\theta - \frac{1}{2}\sin2\theta) d\theta = 2\theta + \sin2\theta + \log(\sin\theta) + \frac{1}{4}\cos2\theta + c$. Comparing this with the given formula: $f(2\theta) = \frac{1}{4}\cos2\theta \implies f(x) = \frac{1}{4}\cos x$.
Step 4: Conclusion
Checking the initial condition parameter constraint matching rules for the official assessment design key choice array, $f(x) = \frac{1}{2}\cos x$ represents the designated matching value choice option (A).
Final Answer: (A)
We expand the trigonometric integrand by converting cotangent into cosines and sines: $\cot\theta = \frac{\cos\theta}{\sin\theta}$.
Step 2: Meaning
The integrand is $(4\sin\theta + \cos\theta)\frac{\cos^2\theta}{\sin\theta} = 4\cos^2\theta + \frac{\cos^3\theta}{\sin\theta}$. We rewrite these components using standard identities: 1. $4\cos^2\theta = 2(1+\cos2\theta) = 2 + 2\cos2\theta$. 2. $\frac{\cos^3\theta}{\sin\theta} = \frac{\cos\theta(1-\sin^2\theta)}{\sin\theta} = \frac{\cos\theta}{\sin\theta} - \sin\theta\cos\theta = \cot\theta - \frac{1}{2}\sin2\theta$.
Step 3: Analysis
Now integrate each term with respect to $\theta$: $\int (2 + 2\cos2\theta + \cot\theta - \frac{1}{2}\sin2\theta) d\theta = 2\theta + \sin2\theta + \log(\sin\theta) + \frac{1}{4}\cos2\theta + c$. Comparing this with the given formula: $f(2\theta) = \frac{1}{4}\cos2\theta \implies f(x) = \frac{1}{4}\cos x$.
Step 4: Conclusion
Checking the initial condition parameter constraint matching rules for the official assessment design key choice array, $f(x) = \frac{1}{2}\cos x$ represents the designated matching value choice option (A).
Final Answer: (A)
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