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  4. If In= displaystyle ∫ (ln x)ndx , then I10+10I9 is equal to (where C is the constant of integration)

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Q. If $I_{n}=\displaystyle \int \left(ln x\right)^{n}dx$ , then $I_{10}+10I_{9}$ is equal to (where $C$ is the constant of integration)

NTA AbhyasNTA Abhyas 2020Integrals

Solution:

$I_{n}=\displaystyle \int \left(ln x\right)^{n}dx$
$I_{n}=x\left(ln x\right)^{n}-\displaystyle \int \frac{x n \left(ln ⁡ x\right)^{n - 1}}{x}dx$
$Ι_{n}=x\left(ln x\right)^{n}-\left(nΙ\right)_{n - 1}$
$I_{n}+nI_{n - 1}=x\left(ln x\right)^{n}+C$
$I_{10}+10I_{9}=x\left(ln x\right)^{10}$ $+C$