$\int x^x\left(1+\log _e x\right) d x$ i

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Indefinite Integration

Question:

$\int x^x\left(1+\log _e x\right) d x$ is equal to

Options:

$x^x \log _e x+C$

$e^{x^x}+C$

$x^x+C$

none of these

Correct Answer:

$x^x+C$

Explanation:

The correct answer is Option (3) → $x^x+C$

$I=\int x^x\left(1+\log x\right) d x$

and,

$x^x=e^{x\log x}$

$⇒I=\int e^{x\log x}(1+\log x)dx$ ...(1)

let,

$t=x\log x$

$⇒\frac{dt}{dx}=\log x+1$

Substituting this in (1),

$I=\int e^tdt$

$=e^t=e^{x\log x}+C$

$=x^x+C$