Integrals Involving Roots — Question 10

Question 10

Use one substitution to evaluate: $$\int\frac{x+1}{\sqrt{x^2+2x+5}}\,dx.$$

Question 10 – Solution

Step 1: Substitute the expression inside the square root. $$u=x^2+2x+5.$$ Differentiate: $$du=(2x+2)\,dx=2(x+1)\,dx, \qquad (x+1)\,dx=\frac12du.$$ Step 2: Rewrite the integral. $$\begin{align*} I&=\int\frac{x+1}{\sqrt{x^2+2x+5}}\,dx\\ &=\frac12\int u^{-1/2}\,du. \end{align*}$$ Step 3: Integrate and return to $x$. $$\begin{align*} I&=\frac12\left(\frac{u^{1/2}}{1/2}\right)+C\\ &=u^{1/2}+C\\ &=\sqrt{x^2+2x+5}+C. \end{align*}$$ $$\boxed{\sqrt{x^2+2x+5}+C}$$