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

1. ganeshie8

substitute $$x=\tan u$$

2. dinamix

i think i s very hard method

3. SolomonZelman

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

4. SolomonZelman

dont forget to substitute back the x.