Let $I(x)=\int\frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If $I(37)-I(24)=\frac14\left(\frac1{b^{\frac1{13}}}-\frac1{c^{\frac1{13}}}\right)$, $b,c\in\mathbf N$, then $3(b+c)$ is equal to

Question

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] With $t=\frac{x-11}{x+15}$, $dt=\frac{26}{(x+15)^2}dx$ and the integrand becomes $t^{-11/13}\frac{dt}{26}$. So $I=\frac1{26}\cdot\frac{13}2t^{2/13}=\frac14t^{2/13}$. $t(37)=\frac12$, $t(24)=\frac13$, so $I(37)-I(24)=\frac14\left(\frac1{4^{1/13}}-\frac1{9^{1/13}}\right)$. $b=4$, $c=9$ and $3(b+c)=39$. Answer (2).

Q. Let $I(x)=\int\frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If $I(37)-I(24)=\frac14\left(\frac1{b^{\frac1{13}}}-\frac1{c^{\frac1{13}}}\right)$, $b,c\in\mathbf N$, then $3(b+c)$ is equal to

Explanation

Reviewed by Dr. Abubacker

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