I have this integral i need to solve, iv been thinking about dividing (x+1)^4 in (x+1)^2 * (x+1)^2 and then dividing the integral but i dont know if this is posible considering that it contains x^1/2 ...can anyone help me figure it out please?
I have this integral i need to solve, iv been thinking about dividing (x+1)^4 in (x+1)^2 * (x+1)^2 and then dividing the integral but i dont know if this is posible considering that it contains x^1/2 ...can anyone help me figure it out please?
$\displaystyle \displaystyle \begin{align*} \int{\frac{1}{\sqrt{x}(x + 1)^4}\,dx} = 2\int{\frac{1}{(x + 1)^4}\,\frac{1}{2\sqrt{x}}\,dx} \end{align*}$
Make the substitution $\displaystyle \displaystyle \begin{align*} u = \sqrt{x} \implies du = \frac{1}{2\sqrt{x}}\,dx \end{align*}$ and the integral becomes $\displaystyle \displaystyle \begin{align*} 2\int{\frac{1}{\left(u^2 + 1\right)^4}\,du} \end{align*}$.
Then make the substitution $\displaystyle \displaystyle \begin{align*} u = \tan{\theta} \implies du = \sec^2{\theta}\,d\theta \end{align*}$