Integral of $\frac{1}{3 x + 3}$

Q: What is the integral of 1 / (3x + 3)?

The integral of (1)/(3 x + 3) is (ln ≤ft(≤ft|3 x + 3|))/(3) + C.

The integral of $\frac{1}{3 x + 3}$ is $\frac{\ln\left(\left|3 x + 3\right|\right)}{3} + C$ .

\int \frac{1}{3 x + 3}\,dx = \frac{\ln\left(\left|3 x + 3\right|\right)}{3} + C

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Graph of f(x) = 1 / (3x + 3) with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 0.23104906.

How to work it out

1
We want an antiderivative of f(x) = 1 / (3x + 3).
$\int \frac{1}{3 x + 3}\,dx$
2
Method: ∫1/(kx+b) dx = ln|kx+b|/k.
3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
$\int \frac{1}{3 x + 3}\,dx = \frac{\ln\left(\left|3 x + 3\right|\right)}{3} + C$
4
Check by differentiating the answer; you must get the original function back:
$\frac{d}{dx}\left[\frac{\ln\left(\left|3 x + 3\right|\right)}{3}\right] = \frac{3 \frac{3 \operatorname{sign}\left(3 x + 3\right)}{\left|3 x + 3\right|}}{9}$
5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 1:
$\int_{0}^{1} \frac{1}{3 x + 3}\,dx = \Big[\frac{\ln\left(\left|3 x + 3\right|\right)}{3}\Big]_{0}^{1} \approx 0.23104906$

LaTeX source of the formulas on this page

\[ \int \frac{1}{3 x + 3}\,dx = \frac{\ln\left(\left|3 x + 3\right|\right)}{3} + C \]
\[ \int \frac{1}{3 x + 3}\,dx \]
\[ \frac{d}{dx}\left[\frac{\ln\left(\left|3 x + 3\right|\right)}{3}\right] = \frac{3 \frac{3 \operatorname{sign}\left(3 x + 3\right)}{\left|3 x + 3\right|}}{9} \]
\[ \int_{0}^{1} \frac{1}{3 x + 3}\,dx = \Big[\frac{\ln\left(\left|3 x + 3\right|\right)}{3}\Big]_{0}^{1} \approx 0.23104906 \]

Definite integrals of the same function

Interval	Value
from 0 to 1	0.23104906
from 0 to 0.5	0.13515504

Questions

What is the integral of 1 / (3x + 3)?

The integral of $\frac{1}{3 x + 3}$ is $\frac{\ln\left(\left|3 x + 3\right|\right)}{3} + C$ .

Why do we add + C?

Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.

What is the integral of 1 / (3x + 3) from 0 to 1?

About 0.23104906.

Integral of 13x+3\frac{1}{3 x + 3}3x+31​

How to work it out

Definite integrals of the same function

Questions

More integrals

Integral of $\frac{1}{3 x + 3}$