# How do you find #\int ( \frac { 1} { 3x + 1} ) d x#?

##### 3 Answers

HI,

Lets call

L=

We will add and subtract **3x** in numerator .

now,

As L is integrtion of two integrals

Lets take ,

X=

Y=

X can be easily solved

X=

X=x+C (C is constant)

Lets solve Y

Now lets take

3x=t

So, diffrentiate

Substitute "dx" in terms of "dt" in Y

Y=

Y=

Y=

Y=

For this integration you must know formula

As **diffrentialtion of (t+1) is 1 this**

Y=

Y=

3x=t

hence

Y=

AS we know

L=X-Y

L=

C+C is Another constant C

L=

The answer is

#### Explanation:

We need

Here,

We perform the substitution

Therefore,

#### Explanation:

In general

here we have

we note that the top isn't quite the bottom differentiated so let us make an adjustment

so the numerator is now the denominator differentiated

so we integrate using the above relationship