Let I(x)= dx(x-11)^(11)/(13)(x+15)^(15)/(13).

Question

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

Accepted Answer

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

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

Solutions · 1