Integral of sqrt(2*x+3) dx

The solution

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  0               
  /               
 |                
 |    _________   
 |  \/ 2*x + 3  dx
 |                
/                 
-2

$$\int\limits_{-2}^{0} \sqrt{2 x + 3}\, dx$$

Integral(sqrt(2*x + 3), (x, -2, 0))

Detail solution

Let $u = 2 x + 3$ .

Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :

$\int \frac{\sqrt{u}}{2}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sqrt{u}\, du = \frac{\int \sqrt{u}\, du}{2}$
  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: $\frac{u^{\frac{3}{2}}}{3}$
Now substitute $u$ back in:

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

  ___   I
\/ 3  + -
        3

$$\sqrt{3} + \frac{i}{3}$$

  ___   I
\/ 3  + -
        3

$$\sqrt{3} + \frac{i}{3}$$

sqrt(3) + i/3

Numerical answer [src]

(1.73139187768924 + 0.333264459583369j)

(1.73139187768924 + 0.333264459583369j)

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

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Integral of sqrt(2*x+3) dx

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The solution