If ∫f(x) sinx cosx d x=(1)/(2(b^2-a^2)) logf(x)+c then f(x) is equal to(A) (1)/(a^2 sin^2 x-b^2

step 1 So this is given in the question. Now we will differentiate both sides. Differentiate both, both

If ∫f(x) sinx cosx d x=(1)/(2(b^2-a^2)) logf(x)+c then f(x)...

If $\int f(x) \sin x \cos x d x=\frac{1}{2\left(b^{2}-a^{2}\right)} \log f(x)+c$
then $f(x)$ is equal to(A) $\frac{1}{a^{2} \sin ^{2} x-b^{2} \cos ^{2} x}$
(B) $\frac{1}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x}$
(C) $\frac{1}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x}$
(D) none of these

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Key Concepts

Separable Differential Equations

A separable differential equation is one in which the variables can be separated onto different sides of the equation, allowing each side to be integrated individually. This method is useful when the derivative of a function can be expressed as a product of a function of the independent variable and a function of the dependent variable.

Trigonometric Integration Techniques

This subject focuses on methods for integrating expressions involving trigonometric functions. It includes the use of trigonometric identities, such as rewriting products like sin x cos x as (1/2)sin 2x, to simplify the integrals and facilitate the integration process.

Differentiation of Integral Equations

This concept involves differentiating both sides of an integral equation to reveal properties or relationships of the integrand. By differentiating an antiderivative expression that is given in terms of an unknown function, one can often derive a differential equation that the function must satisfy.

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