Question

Integrate the function :
f(x) = 1/(x+2)

Answer 1

ANSWER:

To Integrate:

• f(x) = 1/(x +2)

Solution:

We need to find integral of,

$\implies f(x)=\dfrac{1}{x+2}$

That is, we need to find the value of,

$\displaystyle\implies\int\dfrac{1}{x+2}\:\sf dx$

Let us substitute, u = x + 2,

So, differentiating both sides,

$\implies u=x+2$

$\implies\dfrac{du}{dx}=\dfrac{d}{dx}(x+2)$

$\implies\dfrac{du}{dx}=1$

$\implies du=dx$

So,

$\displaystyle\implies\int\dfrac{1}{x+2}\:\sf dx$

$\displaystyle\implies\int\dfrac{1}{u}\:\sf du$

We know that,

$\displaystyle\implies\int\dfrac{1}{t}\:\sf dt=\ln(|t|)+C$

So,

$\displaystyle\implies\int\dfrac{1}{u}\:\sf du$

$\implies \ln(|u|)+C$

Substituting the value of u,

$\implies \ln(|x+2|)+C$

Hence,

$\displaystyle\bf\implies\int\dfrac{1}{x+2}\: dx=ln(|x+2|)+C$