Question ID: #221
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
- (1) 40
- (2) 39
- (3) 22
- (4) 26
Solution:
$$I(x) = \int \frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}} = \int \frac{dx}{\left(\frac{x-11}{x+15}\right)^{\frac{11}{13}} (x+15)^2}$$
Let $\frac{x-11}{x+15} = t \Rightarrow \frac{26}{(x+15)^2}dx = dt \Rightarrow \frac{dx}{(x+15)^2} = \frac{dt}{26}$
$$I(x) = \frac{1}{26} \int \frac{dt}{t^{\frac{11}{13}}} = \frac{1}{26} \cdot \frac{t^{\frac{2}{13}}}{2/13} + C = \frac{1}{4} \left(\frac{x-11}{x+15}\right)^{\frac{2}{13}} + C$$
Now calculate $I(37)$ and $I(24)$:
$$I(37) = \frac{1}{4} \left(\frac{26}{52}\right)^{\frac{2}{13}} = \frac{1}{4} \left(\frac{1}{2}\right)^{\frac{2}{13}} = \frac{1}{4} \cdot \frac{1}{4^{1/13}}$$
$$I(24) = \frac{1}{4} \left(\frac{13}{39}\right)^{\frac{2}{13}} = \frac{1}{4} \left(\frac{1}{3}\right)^{\frac{2}{13}} = \frac{1}{4} \cdot \frac{1}{9^{1/13}}$$
$$I(37) – I(24) = \frac{1}{4} \left(\frac{1}{4^{1/13}} – \frac{1}{9^{1/13}}\right)$$
Comparing with given form: $b=4, c=9$.
$$3(b+c) = 3(4+9) = 3(13) = 39$$
Ans. (2)
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