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JEE Main · 2023 · 01 Feb 2023 · Shift 1 · Indefinite Integrals

If

\int_{0}^{1} (x^{21} + x^{14} + x^{7}) (2 x^{14} + 3 x^{7} + 6)^{1/7} d x = \frac{1}{l} (11)^{m / n}

where

l, m, n \in N

,

m

and

n

are coprime then

l + m + n

is equal to ____.

Answer: 63

Solution

Setting

t = 2 x^{21} + 3 x^{14} + 6 x^{7} = x^{7} (2 x^{14} + 3 x^{7} + 6)

, the radical becomes

(\frac{t}{x ^{7}})^{1/7} = \frac{t ^{1/7}}{x}

, and

d t = 42 x^{6} (x^{14} + x^{7} + 1) d x

. The integrand

x^{7} (x^{14} + x^{7} + 1) \cdot \frac{t ^{1/7}}{x}

is exactly

\frac{1}{42} t^{1/7} d t

, with

t

running from

0

to

11

. Thus the integral equals

\frac{1}{42} \cdot \frac{7}{8} (11)^{8/7} = \frac{1}{48} (11)^{8/7}

, so

l = 48, m = 8, n = 7

and

l + m + n = 63

.

JEE Main · 2023 · 01 Feb 2023 · Shift 1 · Integer answer

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