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Math Help - Differentiation and integration

  1. #1
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    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!!
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  2. #2
    o_O
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    #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) ...
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