Math, asked by sajan6491, 8 hours ago

 \huge \displaystyle \sf \int^{1}_{0} \frac{ {x}^{ {e}^{ \pi } - 1 }  -  {x}^{ {e}^{ \gamma } - 1 } }{ ln( \sqrt[2020]{x} ) }  \: dx

Answers

Answered by chandrakalakavitha21
0

Answer:

Here's slightly different approach

We have

I=∫10−lnx−−−−−√dx=∫10[ln1x]1/2dx

By using IBP

[ln1x]1/2=u⟺du=−12x[ln1x]−1/2dx

dx=dv⟺x=v

I=x[ln1x]1/2∣∣∣10+12∫10[ln1x]−1/2dx=12∫10[ln1x]−1/2dx

Now by substututing

t2=ln1x⟹e−t2=x⟺dx=−2te−t2dt

We get

I=∫∞0e−t2dt=π−−√2

Hence,

∫10−lnx−−−−−√dx=π−−√2

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