U-Substitution for two problems?

Question

I am having issues with the u-substitution of two problems assigned to me. They are Sin(x)/(1+cos^2(x)) and x/(1+x^4). Help with either would be greatly appreciated as I have struggled with both for quite a while

Answer 1

Are you trying to integrate , differentiate, or what?

Answer 2

∫Sin(x)/(1+cos²(x)) dx = ∫Sin(x) dx/(1+cos(x))²  because cos²(x) = [cos (x)]²

Let u = 1+ cos(x)  

du = -sinx dx

-du = sinx dx

- ∫du/u² = - ∫u^-2 du = -(-1)u^-1 +C = 1/u + C



so ∫Sin(x)/(1+cos²(x)) dx = 1/[1+cos(x)] + C

tan^-1(x) = ∫1/1+x² dx

∫x/(1+x^4)dx = ∫x/(1+(x²)²dx

let u = x²

du = 2x dx

(1/2)du = xdx

∫x dx /(1+(x²)² = (1/2) ∫du /(1+u²) = (1/2)tan^-1(u) + C = (1/2)tan^-1(x²) + C

Answer 3

I hope you're talking about Integration.

Integral of sinx/(1+cos^2x) dx=f

Let u = cosx

      du=-sinx dx

     -du=sinx dx

   so

f=-integral du/(1+u^2)

f=-arctanu + C

f=-arctan(cosx)+C

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integral of x/(1+x^4)dx=f

Let u=x^2

du/2=xdx

f=1/2 times integral du/(1+u^2)

f=.5 arctan (u^2) +C

f=.5arctan(x^4)+C

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