Inverse trigonometric functions, L’Hopitals Rule and Integration by Parts

Please see the attached file for the fully formatted problems.

1. Prove that d/dx (csc^-1 x) = – (1 / x(square root x^2 – 1)

2. Find the derivative of the function. Simplify where possible.

y = tan^-1 (x/Q) + ln (square root (x-Q)/(x+Q))

3. Find the limit using l’Hopital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hopital’s Rule does not apply, explain why.

lim x-> infinity (x – ln x)

4. Use a graph to estimate the value of the limit. Then use l’Hopital’s Rule to find the exact value.

lim x -> pi/4 (tan X)^(tan2x)

5. Evaluate the integral

integral x^2 cos mx dx