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!!
#1:
$\displaystyle \tan y = x^{2}$
$\displaystyle \sec^{2}(y) \cdot y' = 2x$
Simply solve for y' (it's ok to have dy/dx in terms of both x and y)
#2
$\displaystyle \int \frac{x}{4x^4 + 1} \: dx \: \: = \: \: \int \frac{x}{(2x)^{2} + 1} \: dx$
Use: $\displaystyle u = 2x^{2} \: \: \Rightarrow \: \: du = 4x dx \: \: \iff \: \: x dx = \frac{du}{4}$
So: $\displaystyle \int \frac{xdx}{(2x)^2 + 1} \: dx = \frac{1}{4} \int \frac{du}{u^{2} + 1}$
This should remind you of arctan(u) ...