If ∫dxx+x7=p(x) then ∫x6x+x7 dx=___

∵p(x)=∫dxx+x7 ∫x6x(1+x6)dx =∫(x6+1)−1x(1+x6)dx =log|x|−p(x)+c

If ∫dxx+x7=p(x) then ∫x6x+x7 dx=___

∵p(x)=∫dxx+x7 ∫x6x(1+x6)dx =∫(x6+1)−1x(1+x6)dx =log|x|−p(x)+c

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If $\int \frac{d x}{x + x^{7}} = p (x)$ then $\int \frac{x^{6}}{x + x^{7}} d x =___$

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$\log | x | + p (x) + c$

$x + p (x) + c$

$x - p (x) + c$

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

answer is A.

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Detailed Solution

$∵ p (x) = \int \frac{d x}{x + x^{7}} \int \frac{x^{6}}{x (1 + x^{6})} d x = \int \frac{(x^{6} + 1) - 1}{x (1 + x^{6})} d x = \log | x | - p (x) + c$

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If $\int \frac{d x}{x + x^{7}} = p (x)$ then $\int \frac{x^{6}}{x + x^{7}} d x =___$