Question

what is primitive this function f(x)= 1/(x^2+1)^3..

Answer 1

substitute \(x=\tan u\)

Answer 2

i think i s very hard method

Answer 3

No, it should be quite easy knowing the identities.

\( \displaystyle\large \int\limits_{ }^{ }\frac{1}{(x^2+1)^3} dx\)

\(u=\tan(x)\) is a standard trig substitution.
then, \(du=(\sec^2u)du\)

\( \displaystyle\large \int\limits_{ }^{ }\frac{\sec^2u}{(\tan^2u+1)^3} du\)

\( \displaystyle\large \int\limits_{ }^{ }\frac{\sec^2u}{(\sec^2u)^3} du\)

\( \displaystyle\large \int\limits_{ }^{ }(\cos^2u) du\)

then you know that:
cos(2w)=cos²w-sin²w
cos(2w)=2cos²w-1
cos(2w)+1=2cos²w
½(cos(2w)+1)=cos²w

so it follows that:

\( \displaystyle\large \large \frac{ 1}{ 2} \int\limits_{ }^{ }
\large \left(\cos(2u)+1\right) du\)

Answer 4

dont forget to substitute back the x.