If int frac{dx}{x + x^7} = p(x),then int frac | vedclass

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(A) We are given that $\int \frac{dx}{x + x^7} = p(x)$.Consider the integral $I = \int \frac{x^6}{x + x^7} dx$.We can rewrite the integrand as:$I = \i

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$\ln |x| - p(x) + c$

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(A) We are given that $\int \frac{dx}{x + x^7} = p(x)$.Consider the integral $I = \int \frac{x^6}{x + x^7} dx$.We can rewrite the integrand as:$I = \int \frac{x^6}{x(1 + x^6)} dx = \int \frac{x^5}{1 + x^6} dx$.Alternatively,we can manipulate the original expression:$I = \int \frac{x^6}{x(1 + x^6)} dx = \int \frac{(1 + x^6) - 1}{x(1 + x^6)} dx$.Splitting the integral,we get:$I = \int \frac{1 + x^6}{x(1 + x^6)} dx - \int \frac{1}{x(1 + x^6)} dx$.$I = \int \frac{1}{x} dx - \int \frac{1}{x + x^7} dx$.Since $\int \frac{1}{x} dx = \ln |x| + c_1$ and $\int \frac{1}{x + x^7} dx = p(x) + c_2$,we have:$I = \ln |x| - p(x) + c$.

If $\int \frac{dx}{x + x^7} = p(x)$,then $\int \frac{x^6}{x + x^7} dx$ is equal to

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