Let I(x) = int frac{dx}{(x-11)^{frac{11}{13}} | vedclass

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Question

(B) Given $I(x) = \int \frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$.Rewrite the integrand as $I(x) = \int \left( \frac{x-11}{x+15} \right)

Vedclass · Accepted Answer

$39$

Vedclass · Answer

(B) Given $I(x) = \int \frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$.Rewrite the integrand as $I(x) = \int \left( \frac{x-11}{x+15} \right)^{-\frac{11}{13}} \cdot \frac{1}{(x+15)^2} dx$.Let $t = \frac{x-11}{x+15}$. Then $dt = \frac{(x+15) - (x-11)}{(x+15)^2} dx = \frac{26}{(x+15)^2} dx$.Thus,$I(x) = \frac{1}{26} \int t^{-\frac{11}{13}} dt = \frac{1}{26} \cdot \frac{t^{\frac{2}{13}}}{\frac{2}{13}} + C = \frac{1}{4} \left( \frac{x-11}{x+15} \right)^{\frac{2}{13}} + C$.Now,$I(37) - I(24) = \frac{1}{4} \left( \frac{37-11}{37+15} \right)^{\frac{2}{13}} - \frac{1}{4} \left( \frac{24-11}{24+15} \right)^{\frac{2}{13}}$.$= \frac{1}{4} \left( \frac{26}{52} \right)^{\frac{2}{13}} - \frac{1}{4} \left( \frac{13}{39} \right)^{\frac{2}{13}} = \frac{1}{4} \left( \frac{1}{2} \right)^{\frac{2}{13}} - \frac{1}{4} \left( \frac{1}{3} \right)^{\frac{2}{13}}$.$= \frac{1}{4} \left( \frac{1}{4^{\frac{1}{13}}} - \frac{1}{9^{\frac{1}{13}}} \right)$.Comparing with the given form,$b = 4$ and $c = 9$.Therefore,$3(b+c) = 3(4+9) = 3(13) = 39$.

Let $I(x) = \int \frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If $I(37) - I(24) = \frac{1}{4} \left( \frac{1}{b^{\frac{1}{13}}} - \frac{1}{c^{\frac{1}{13}}} \right)$,where $b, c \in \mathbb{N}$,then $3(b+c)$ is equal to

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