# Test-2

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*(This question paper consists of 6 questions. All questions carry 5 marks each.)*

*(Attempt any 5 questions.)*

- Test for convergence of the integral [math]\int_0^1x^p(log\frac{1}{x})^q~dx[/math]
- Show that [[Image:]]exists iff n < m+1
- Discuss the convergence of [[Image:]]
- State Green's theorem and verify it for I = [[Image:]]; where C is the closed curve formed by y
^{3}= x^{2}and y=x between (0,0) and (1,1). - Change the order of integration in the inegral[[Image:]]and evaluate it.
- Evaluate[[Image:]]taken over the upper half of the circle x
^{2}+ y^{2}= 2ax. - State and prove Abel's Test for the convergence of Improper Integrals.