Mathematics - Indefinite Integration Question with Solution | TestHub

$\begin{array}{rl}& I(x)=\int \frac{dx}{(x-11{)}^{\frac{11}{13}}(x+15{)}^{\frac{15}{13}}}\\ & \text{ Put }\frac{x-11}{x+15}=t\Rightarrow \frac{26}{(x+5{)}^2}dx=dt\\ & I(x)=\frac{1}{26}\int \frac{dt}{t^{1/13}}=\frac{1}{26}\cdot \frac{t^{2/13}}{2/13}\end{array}$


$\begin{array}{rl}& \mathrm{I}(\mathrm{x})=\frac{1}{4}{\left(\frac{\mathrm{x}-11}{\mathrm{x}+15}\right)}^{2/13}+\mathrm{C}\\ & \mathrm{I}(37)-\mathrm{I}(24)=\frac{1}{4}{\left(\frac{26}{52}\right)}^{2/13}-\frac{1}{4}{\left(\frac{13}{39}\right)}^{2/13}\\ & =\frac{1}{4}\left(\frac{1}{2^{2/13}}-\frac{1}{3^{2/13}}\right)\\ & =\frac{1}{4}\left(\frac{1}{4^{1/13}}-\frac{1}{9^{1/13}}\right)\\ & \therefore \mathrm{b}=4,\mathrm{c}=9\\ & 3(\mathrm{b}+\mathrm{c})=39\end{array}$