Mathematics

Evaluate : $$\displaystyle\int {{x^x}.\left( {1 + \log x} \right)} \,dx$$ is equal to,


ANSWER

$${x^x} + C$$


SOLUTION

$$\int {{x^x}\left( {1 + \log x} \right)} dx$$

Let $${x^x} = t$$

Applying $$log$$ on both the side

  $$log \ x^{x}=log \ t$$ 

$$x\log x = \log t$$

Taking Derivative w.r.t. $$x$$

$$\left( {x \times \dfrac {1}{x}+ \log x} \right)dx = \dfrac {1}{t}dt$$

$$ \Rightarrow \left( {1 + \log x} \right) \times {x^x}dx = dt$$......$$\left ( \because t=x^{x} \right )$$

$$ \Rightarrow \int {dt} $$

$$ = t + c$$

$$ = {x^x} + c$$

Hence option $$C$$ is the answer.

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Single Correct Medium Published on 17th 09, 2020
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