Verified Solution • 2025–26 Session moderate

QJEEM23Ma96e7: JEE Main 2023 Mathematics - Class 12 JEEM23Ma96e7 Solution: If ∫ 0 1 ( x 21 + x 14 + x 7 ) ( 2 x 14 + 3 x 7 + 6 ) 1 / 7 d x …

If ∫01(x21+x14+x7)(2x14+3x7+6)1/7dx=1l(11)m/n where l,m,n∈N,ml, m, n \in \mathbb{N}, m and nn are coprime then l+m+nl+m+n is equal to ____________.

Step-by-Step Verified Solution

I=∫01(x21+x14+x7)(2x14+3x7+6)1/7dxI=\int_{0}^{1}\left(x^{21}+x^{14}+x^{7}\right)\left(2 x^{14}+3 x^{7}+6\right)^{1 / 7} d x

I=∫01(x20+x13+x6)(2x21+3x14+6x7)1/7dxI=\int_{0}^{1}\left(x^{20}+x^{13}+x^{6}\right)\left(2 x^{21}+3 x^{14}+6 x^{7}\right)^{1 / 7} d x

Let 2x21+3x14+6x7=t2 x^{21}+3 x^{14}+6 x^{7}=t

⇒42(x20+x13+x6)dx=dt\Rightarrow 42\left(x^{20}+x^{13}+x^{6}\right) d x=d t

I=142∫011t1/7dt=14278[t8/7]011I=\frac{1}{42} \int_{0}^{11} t^{1 / 7} d t=\frac{1}{42} \frac{7}{8}\left[t^{8 / 7}\right]_{0}^{11}

=148(11)8/7=\frac{1}{48} (11)^{8/7}

∴I=48,m=8,n=7\therefore \quad I=48, m=8, n=7

∴l+m+n=63\therefore \quad l+m+n=63

Related Learning Resources & Exam Preparation

JEE Main PYQ Question Bank
Browse year-wise authentic previous years' questions with solutions
Authentic CBT Mock Tests
Simulate official exam environment with live timer and percentile scoring