If $I_{n}=\int(\ln x)^n d x$ then $I_n+n

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Target Exam

CUET

Subject

-- Mathematics - Section A

Chapter

Indefinite Integration

Question:

If $I_{n}=\int(\ln x)^n d x$ then $I_n+n I_{n-1}$ is equal to

Options:

$x(\ln x)^{n-1}$

$x(\ln x)^n$

$n x(\ln x)^n$

none of these

Correct Answer:

$x(\ln x)^n$

Explanation:

$I_{n}=\int(\ln x)^n d x$

$I_n=x(\ln x)^n-n \int(\ln x)^{n-1} d x$

$I_n=x(\ln x)^n-n I_{n-1}$

$I_n+n I_{n-1}=x(\ln x)^n$

Hence (2) is the correct answer.