Question

$\red{\displaystyle \int_{0}^{1} \sqrt{ \sqrt{ \ln( \rm{{x) }}} } }$​

Answer 1

Answer:

The function is not defined in given limits because in range 0 < x < 1

$\red{ln(x) < 0 \implies \: \sqrt{ ln(x) } \: \: = Imaginary}$

Answer 2

$\red{\displaystyle \int_{0}^{1} \sqrt{ \sqrt{ \ln( \rm{{x) }}} } }$

$\red{\displaystyle \rm{\int_{ \infty }^{ 0 } \sqrt{ \sqrt{ ( \rm{{ - u) }}} } \cdot - {e}^{ - u} \cdot du}}$

$\red{\displaystyle \rm{\int_{ \infty }^{ 0 } { { ( \rm{{ - u)^{ \frac{1}{4} } }}} } \cdot {e}^{ - u} \cdot du}}$

$\red{\displaystyle \rm{( - 1) {}^{ \frac{1}{4} } \int_{ 0 }^{ \infty } { { \rm{{ }}} } {e}^{ - u} \cdot {u}^{ \frac{1}{4} } \cdot du}}$

$\red{\displaystyle \rm{( - 1) {}^{ \frac{1}{4} } \int_{ 0 }^{ \infty } { { \rm{{ }}} } {e}^{ - u} \cdot {u}^{ \frac{5}{4} - 1 } \cdot du}}$

$\large\rm \red{ {e}^{i \frac{\pi}{4} } \cdot \Gamma( \frac{5}{4}) }$

$\large\rm \red { {} \bigg({\frac{1}{ \sqrt{2} } + i \frac{1}{ \sqrt{2} } \bigg) } \cdot \Gamma( \frac{5}{4}) }$

$\large\rm \red{ \frac{1}{ \sqrt{2} }(1 + i) \cdot \Gamma( \frac{5}{4} )}$