U6FM Homework 1 Due Wed 30.9.15

Homework on paper. 27 MARKS

1. Find the values of $x$ for which $2\sinh x + 3\cosh x = 4$

[3 marks]

2. Find, simplifying your answer as much as possible!

1. $$\int_{0}^{2} \dfrac {2x^3}{1-x^2} dx$$
2. $$\int_{0}^{2} \dfrac {2x}{1-x^2} dx$$
3. $$\int_0^2 \dfrac {2x}{\sqrt {1-x^2}} dx$$
4. $$\int_0^2 \dfrac {2}{\sqrt {1-x^2}} dx$$

   Note that the last 2 parts have complex (not real) solutions. Bonus points if you can state what they are!! In fact the last one is $$-\pi(1+2n)-2i\ln{ (2 \pm \sqrt3)}$$

[12 marks]

3. Find $$\int \text{sech}^2 x\tanh^7 x dx$$

[4 marks]

4. Find $\dfrac{d}{dx} (\text{sech} x)$ from the definitions

[4 marks]

5. Find $$\int_1^2 \dfrac {1}{\sinh x + 2 \cosh x} dx$$

[4 marks]