Question

Integrate: log(ax+b)/(ax+b) dx​

Answer 1

`(1)/(2a) \bigg[log(ax + b) \bigg]^(2) + c`

Explanation:

`\int (log(ax + b))/((ax + b)) dx \\ \\ let \: log(ax + b) = t \\ \\ (1)/((ax + b)) .a \: dx = dt \\ \\ (1)/((ax + b)) \: dx = (1)/(a) dt \\ \\ \therefore \: \int (log(ax + b))/((ax + b)) dx = \int t. (1)/(a) dt \\ \\ = (1)/(a) \int t. dt \\ \\ = (1)/(a) * \frac{ {t}^(2) }{2} + c \\ \\ = (1)/(2a) \bigg[log(ax + b) \bigg]^(2) + c \\ \\ \purple{ \bold{\therefore \: \int (log(ax + b))/((ax + b)) dx = (1)/(2a) \bigg[log(ax + b) \bigg]^(2) + c}}`