Q.Find the integrals of the following:

(i) Let $I=\displaystyle\int\frac{dx}{6x-7-x^2}$.

Step 1. $6x-7-x^2=-(x^2-6x+7)=-\big[(x-3)^2-9+7\big]=-\big[(x-3)^2-2\big]=2-(x-3)^2$.

Step 2. This is Type I, $a^2-u^2$, with $a=\sqrt2$, $u=x-3$: $I=\dfrac1{2\sqrt2}\log\left|\dfrac{\sqrt2+(x-3)}{\sqrt2-(x-3)}\right|+c$.

(ii) Let $I=\displaystyle\int\frac{dx}{(x+1)^2-25}$.

Step 1. Already in Type I form $u^2-a^2$ with $u=x+1$, $a=5$.

Step 2. $I=\dfrac1{2(5)}\log\left|\dfrac{(x+1)-5}{(x+1)+5}\right|+c=\dfrac1{10}\log\left|\dfrac{x-4}{x+6}\right|+c$.

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

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

• Sign error completing the square when the leading coefficient of $x^2$ is negative. …