Question

Can I get an answer to this question please c

Answer 1

Answer::

`(\ln |x|)/(2)+\frac{\ln |x^{}+6|}{2}+C`

Explanation:

Given:

`I=\int (x+3)/(x^(2)+6 x)dx`
`\begin{gathered} \text{Let }u=x^2+6x \\ \implies(du)/(dx)=2x+6=2(x+3) \\ \implies dx=(du)/(2(x+3)) \end{gathered}`

Substitute u and dx into I.

`\begin{gathered} I=\int (x+3)/(x^2+6x)dx \\ \implies I=\int (x+3)/(u)(du)/(2(x+3))=\int (1)/(u)(du)/(2)=(1)/(2)\int (1)/(u)du \end{gathered}`

The integral of 1/u is the standard integral ln |u|.

Therefore:

`I=(1)/(2)\ln |u|+C,C\text{ a constant of integration}`

SUbstitute back the expression for u.

`\begin{gathered} I=(1)/(2)\ln |x^2+6x|+C=\frac{\ln|x||x^{}+6|}{2}+C \\ =(\ln|x|)/(2)+(\ln |x+6|)/(2)+C \end{gathered}`

The result of the given integral is:

`(\ln |x|)/(2)+\frac{\ln |x^{}+6|}{2}+C`