f(x) = int frac{dx}{sin^6 x} is a polynomial | vedclass

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(A) We have $f(x) = \int \csc^6 x \, dx$.Using the identity $\csc^2 x = 1 + \cot^2 x$,we can write:$f(x) = \int \csc^4 x \cdot \csc^2 x \, dx = \int (

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$5$ in $\cot x$

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(A) We have $f(x) = \int \csc^6 x \, dx$.Using the identity $\csc^2 x = 1 + \cot^2 x$,we can write:$f(x) = \int \csc^4 x \cdot \csc^2 x \, dx = \int (1 + \cot^2 x)^2 \csc^2 x \, dx$.Let $u = \cot x$,then $du = -\csc^2 x \, dx$,which implies $\csc^2 x \, dx = -du$.Substituting these into the integral:$f(x) = \int (1 + u^2)^2 (-du) = -\int (1 + 2u^2 + u^4) \, du$.Integrating with respect to $u$:$f(x) = -(u + \frac{2u^3}{3} + \frac{u^5}{5}) + C$.Substituting $u = \cot x$ back:$f(x) = -\cot x - \frac{2}{3} \cot^3 x - \frac{1}{5} \cot^5 x + C$.This is a polynomial in $\cot x$ of degree $5$.

$f(x) = \int \frac{dx}{\sin^6 x}$ is a polynomial of degree

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