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find antiderivative

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find antiderivative

find antiderivative of a function f (x) = x ^ 6 / (1 + x ^ 12)


i know result but i donot know how to solve this problem step by step

i want to know that how is the solving step by step

I will at least get you started then.

You need to factor the denominator into quadratic factors and then convert to partial fractions. The rest is pretty much paint by numbers math. To get you started

x¹² + 1 = (x² + √2 x + 1)*(x² - √2 x + 1)*
(x² + ((√3 - 1)/√2)x + 1)*(x² - ((√3 - 1)/√2)x + 1)*
((x² + ((√3 + 1)/√2)x + 1)*(x² - ((√3 + 1)/√2)x + 1)

How did I do that? Use
(x² + ax + 1)((x² - ax + 1) = x^4 + (2 - a²)x² + 1)

I will at least get you started then.

You need to factor the denominator into quadratic factors and then convert to partial fractions. The rest is pretty much paint by numbers math. To get you started

x^12 + 1 = (x^2 + Ax + 1)*(x^2 - Ax + 1)*
(x^2 + Bx + 1)*(x^2 - Bx + 1)*
(x^2 + Cx + 1)*(x^2 - Cx + 1)

Where A = sqrt(2)
B = (sqrt(ˆš3) - 1)/sqrt(2)
C = (sqrt(ˆš3) + 1)/sqrt(2)

How did I do that? Use
(x^2n + ax^n + 1)((x^2n - ax^n + 1) = x^4n + (2 - a^2)x^2n + 1)

thanks a lot. let me consider.

thank you very much! i am afraid that this function under cannot regard that (u.du)/(1+u^2) because dx is not equal to du (du=6x^5.dx)
Please help me to solve this problem. thank you a lot

Hi,

you are obviously right, I did checked the $du$, $dx$ part.
I shall chek it again ........ and see what can be done

Sorry

Let $u = x^{6}$ then the function under consideration assumes the form
$\hat{f}(u) = \frac{u}{1+u^{2}}$ and it is easy to get that an antiderivative of $\hat{f}$ is given by the
$\hat{F}(u) = \frac{1}{2}ln (1 + u^2) + c$, $c$ a constant.

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