Question

If \(\int f(x) \sin x \cos x d x=\frac{1}{2\left(b^{2}-a^{2}\right)} \log f(x)+c\), where \(c\) is the constant of integration, then \(f(x)=\)
(A) \(\frac{2}{\left(b^{2}-a^{2}\right) \sin 2 x}\)
(B) \(\frac{2}{a b \sin 2 x}\)
(C) \(\frac{2}{\left(b^{2}-a^{2}\right) \cos 2 x}\)
(D) \(\frac{2}{a b \cos 2 x}\)

Answer 1

Ans: (C)
Hint \(: \frac{d}{d x}\left[\frac{1}{2\left(b^{2}-a^{2}\right)} \log f(x)+c\right]\)
\(=\frac{f^{\prime}(x)}{f(x)} \times \frac{1}{2\left(b^{2}-a^{2}\right)}=f(x) \sin x \cos x\) (by question)
\(\Rightarrow y^{2} \sin 2 x=\frac{d y}{d x} \frac{1}{b^{2}-a^{2}} \quad(\) Take \(f(x)=y)\)
\(\Rightarrow \int \frac{d y}{y^{2}}=\left(b^{2}-a^{2}\right) \int \sin 2 x d x \Rightarrow y=\frac{2}{\left(b^{2}-a^{2}\right) \cos 2 x}\)

Answer 2

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Answer 3

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