Question

Integrate the following: \[ \int \frac{d y}{y^{2}+6 y+10} \]

Answer 1

`\mathrm{tan}^(-1)(y+3)+c`

Note:

`\int (dx)/(x^2+a^2)=(1)/(a)\mathrm{tan^(-1)}((x)/(a))+c`

Explanation:

`\[ \int (d y)/(y^(2)+6 y+10) \]\\\\=\int (dy)/(y^2+6y+9+1)\\\\= \int (dy)/((y+3)^2+1^2)\\\\=(1)/(1)\mathrm{tan}^(-1)((y+3)/(1))+c\\=\mathrm{tan}^(-1)(y+3)+c`