Q.Find the integrals of the following:

(i) Let $I=\displaystyle\int\frac{dx}{\sqrt{(2+x)^2-1}}$.

Step 1. Already Type I, $u^2-a^2$ under a root, with $u=x+2$, $a=1$; note $(x+2)^2-1=x^2+4x+3$.

Step 2. $I=\log\left|(x+2)+\sqrt{(x+2)^2-1}\right|+c=\log\left|(x+2)+\sqrt{x^2+4x+3}\right|+c$.

(ii) Let $I=\displaystyle\int\frac{dx}{\sqrt{x^2-4x+5}}$.

Step 1. $x^2-4x+5=(x-2)^2-4+5=(x-2)^2+1$.

Step 2. Type I, $\displaystyle\int\frac{du}{\sqrt{u^2+a^2}}=\log|u+\sqrt{u^2+a^2}|+c$, with $u=x-2$, $a=1$: $I=\log\left|(x-2)+\sqrt{x^2-4x+5}\right|+c$.

(iii) Let $I=\displaystyle\int\frac{dx}{\sqrt{9+8x-x^2}}$. …

Complete the square to reduce a quadratic surd to Type I, then app …

• Misreading $9+8x-x^2$ as an $x^2-a^2$ form instead of $a^2-x^2$ -- the negative leading coefficient decides which formula applies. …