Integral of sqrt(3x+1) dx

The solution

You have entered [src]

  1               
  /               
 |                
 |    _________   
 |  \/ 3*x + 1  dx
 |                
/                 
0

$$\int\limits_{0}^{1} \sqrt{3 x + 1}\, dx$$

Integral(sqrt(3*x + 1), (x, 0, 1))

Detail solution

Let $u = 3 x + 1$ .

Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :

$\int \frac{\sqrt{u}}{3}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sqrt{u}\, du = \frac{\int \sqrt{u}\, du}{3}$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}$
  So, the result is: $\frac{2 u^{\frac{3}{2}}}{9}$
Now substitute $u$ back in:

$\frac{2 \left(3 x + 1\right)^{\frac{3}{2}}}{9}$
Now simplify:

$\frac{2 \left(3 x + 1\right)^{\frac{3}{2}}}{9}$
Add the constant of integration:

$\frac{2 \left(3 x + 1\right)^{\frac{3}{2}}}{9}+ \mathrm{constant}$

The answer is:

$\frac{2 \left(3 x + 1\right)^{\frac{3}{2}}}{9}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                                 3/2
 |   _________          2*(3*x + 1)   
 | \/ 3*x + 1  dx = C + --------------
 |                            9       
/

$$\int \sqrt{3 x + 1}\, dx = C + \frac{2 \left(3 x + 1\right)^{\frac{3}{2}}}{9}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

14/9

$$\frac{14}{9}$$

14/9

$$\frac{14}{9}$$

14/9

Numerical answer [src]

1.55555555555556

1.55555555555556

The graph

Use the examples entering the upper and lower limits of integration.

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of sqrt(3x+1) dx

Limits of integration:

The graph:

Piecewise:

The solution