If $$\int_\limits{0}^{1}\left(x^{21}+x^{14}+x^{7}\right)\left(2 x^{14}+3 x^{7}+6\right)^{1 / 7} d x=\frac{1}{l}(11)^{m / n}$$ where $l, m, n \in \mathbb{N}, m$ and $n$ are coprime then $l+m+n$ is equal to ____________.
Answer (integer)
63
Solution
$I=\int_{0}^{1}\left(x^{21}+x^{14}+x^{7}\right)\left(2 x^{14}+3 x^{7}+6\right)^{1 / 7} d x$ <br/><br/>$I=\int_{0}^{1}\left(x^{20}+x^{13}+x^{6}\right)\left(2 x^{21}+3 x^{14}+6 x^{7}\right)^{1 / 7} d x$
<br/><br/>Let $2 x^{21}+3 x^{14}+6 x^{7}=t$
<br/><br/>$\Rightarrow 42\left(x^{20}+x^{13}+x^{6}\right) d x=d t$
<br/><br/>$I=\frac{1}{42} \int_{0}^{11} t^{1 / 7} d t=\frac{1}{42} \frac{7}{8}\left[t^{8 / 7}\right]_{0}^{11}$
<br/><br/>$=\frac{1}{48} (11)^{8/7}$
<br/><br/>$\therefore \quad I=48, m=8, n=7$
<br/><br/>$\therefore \quad l+m+n=63$
About this question
Subject: Mathematics · Chapter: Definite Integration · Topic: Properties of Definite Integrals
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