Mathxyz - upsc.xyz
0 votes
Evaluate: $ I = \int_{0}^{1} \sqrt[3]{x log\left(\frac{1}{x}\right)} dx $

(10 marks)
asked Aug 21, 2017 by randomisation (1,920 points)  

1 Answer

0 votes
Best answer
Let $ x = e^{-t} \implies dx = -e^{-t} dt \\ x = 0 \implies t = \infty \ and\  x = 1 \implies t = 0 $

Thus, $ I = \int_{0}^{\infty} \sqrt[3]{-e^{-t} t}\: e^{-t} \ dt \\  = -\int_{0}^{\infty} t^{1/3} e^{\frac{-4t}{3}} \ dt \\ $

Now let,  $ 4t/3 = u \implies dt = \frac{3}{4} du, \\ \implies I = - \left( \frac{3}{4} \right)^{4/3} \int_{0}^{\infty} u^{1/3} e^{-u} \ dt = - \left( \frac{3}{4} \right)^{4/3} \Gamma(4/3) $
answered Aug 21, 2017 by randomisation (1,920 points)  
selected Aug 21, 2017 by randomisation
Welcome to MathXyz, where you can ask questions and receive answers from other members of the community. Please strictly ask questions from UPSC Mathematics syllabus.
36 questions
26 answers
2 comments
12 users