Solve ∫ dT/T-27=kintdt

$$\int \frac { d T } { T - 27 } = k \int d t$$

Solve for k

$$\left\{\begin{matrix}k=\frac{\ln(|T-27|)+С}{t+С_{1}}\text{, }&t\neq -С\text{ and }T\neq 27\\k\in \mathrm{R}\text{, }&С=-\ln(|T-27|)\text{ and }t=\ln(|T-27|)\text{ and }T\neq 27\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$k\int 1\mathrm{d}t=\int \frac{1}{T-27}\mathrm{d}T$$

The equation is in standard form.

$$\left(t+С\right)k=\ln(|T-27|)+С$$

Divide both sides by t+С.

$$\frac{\left(t+С\right)k}{t+С}=\frac{\ln(|T-27|)+С}{t+С}$$

Dividing by t+С undoes the multiplication by t+С.

$$k=\frac{\ln(|T-27|)+С}{t+С}$$

Divide \ln(|T-27|)+С by t+С.

$$k=\frac{\ln(|T-27|)+С}{t+С_{1}}$$