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JEE Main · 2025 · 23 Jan 2025 · Shift 1 · Indefinite Integrals

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

(A)40
(B)39✓ Correct
(C)22
(D)26

Answer: (B)

Solution

Let

t = \frac{x - 11}{x + 15}

, so

d t = \frac{26}{( x + 15 ) ^{2}} d x

. Writing the integrand as

\frac{1}{( x + 15 ) ^{2}} \cdot t^{- 11/13}

gives

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

I (37) = \frac{1}{4} (\frac{26}{52})^{2/13} = \frac{1}{4} (\frac{1}{2})^{2/13}, I (24) = \frac{1}{4} (\frac{13}{39})^{2/13} = \frac{1}{4} (\frac{1}{3})^{2/13}

I (37) - I (24) = \frac{1}{4} [\frac{1}{2 ^{2/13}} - \frac{1}{3 ^{2/13}}] = \frac{1}{4} [\frac{1}{4 ^{1/13}} - \frac{1}{9 ^{1/13}}]

So

b = 4

,

c = 9

, and

3 (b + c) = 3 (13) = 39

.

JEE Main · 2025 · 23 Jan 2025 · Shift 1 · Single correct option

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