JEE Main 2025 Jan 23 Shift 1, Mathematics Q2: Methods of Integration

Factor out $(x+15{)}^{26/13}$ from the denominator. Write the integrand as $\frac{1}{(x+15{)}^2}{\left(\frac{x+15}{x-11}\right)}^{11/13}$.

Let $t=\frac{x-11}{x+15}$. Then $dt=\frac{26}{(x+15{)}^2}dx$, so

$I(x)=\frac{1}{26}\int t^{-11/13}dt=\frac{1}{26}\cdot \frac{t^{2/13}}{2/13}=\frac{1}{4}{\left(\frac{x-11}{x+15}\right)}^{2/13}+C$.

Evaluating:

$I(37)-I(24)=\frac{1}{4}\left[{\left(\frac{26}{52}\right)}^{2/13}-{\left(\frac{13}{39}\right)}^{2/13}\right]=\frac{1}{4}\left[\frac{1}{2^{2/13}}-\frac{1}{3^{2/13}}\right]=\frac{1}{4}\left[\frac{1}{4^{1/13}}-\frac{1}{9^{1/13}}\right]$.

Hence $b=4$, $c=9$, so $3(b+c)=39$.