1. ## Differentiation and integration

Use implicit differentiation to find dy/dx in terms of x if a tan y = (x^2) where (- pi/2) < y < (pi/2)

Deduce the exact value of ∫ [ x/ (1+4(x^4))] for x from (1/2)^0.5 to [(3^0.5)/2]^0.5

Thank you!!

2. #1:

$\tan y = x^{2}$
$\sec^{2}(y) \cdot y' = 2x$

Simply solve for y' (it's ok to have dy/dx in terms of both x and y)

#2
$\int \frac{x}{4x^4 + 1} \: dx \: \: = \: \: \int \frac{x}{(2x)^{2} + 1} \: dx$

Use: $u = 2x^{2} \: \: \Rightarrow \: \: du = 4x dx \: \: \iff \: \: x dx = \frac{du}{4}$

So: $\int \frac{xdx}{(2x)^2 + 1} \: dx = \frac{1}{4} \int \frac{du}{u^{2} + 1}$

This should remind you of arctan(u) ...