Question

Evaluate : $\int\limits^3_{-2} {\frac{1}{x+5}} \, dx$

Answer 1

let us assume that

$x + 5 = t \\ \\ dx = dt$

$\int\limits^3_{-2} {\frac{1}{x+5}} \, dx = \int\limits^3_{-2} \: \frac{1}{t} dt \\ \\ as \: \: we \: \: know \: \: that \: \int \: \frac{1}{x} dx = log(x) + c \\ \\ \int\limits^3_{-2} = [log \: t] | - 2 \: to\: 3$

redo substitution and apply lower and upper limit

$= log(x + 5) put \: limits \\ \\ = log(2 + 5) - log( - 3 + 5) \\ \\ = log(7) - log(2) \\ \\ \int\limits^3_{-2} {\frac{1}{x+5}} \, dx = log( \frac{7}{2} )$