Integral of sqrt(5-3x) dx

The solution

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  1               
  /               
 |                
 |    _________   
 |  \/ 5 - 3*x  dx
 |                
/                 
0

$$\int\limits_{0}^{1} \sqrt{5 - 3 x}\, dx$$

Integral(sqrt(5 - 3*x), (x, 0, 1))

Detail solution

Let $u = 5 - 3 x$ .

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

$\int \left(- \frac{\sqrt{u}}{3}\right)\, du$
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:

$- \frac{2 \left(5 - 3 x\right)^{\frac{3}{2}}}{9}$
Add the constant of integration:

$- \frac{2 \left(5 - 3 x\right)^{\frac{3}{2}}}{9}+ \mathrm{constant}$

The answer is:

$- \frac{2 \left(5 - 3 x\right)^{\frac{3}{2}}}{9}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                                 3/2
 |   _________          2*(5 - 3*x)   
 | \/ 5 - 3*x  dx = C - --------------
 |                            9       
/

$$\int \sqrt{5 - 3 x}\, dx = C - \frac{2 \left(5 - 3 x\right)^{\frac{3}{2}}}{9}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

      ___        ___
  4*\/ 2    10*\/ 5 
- ------- + --------
     9         9

$$- \frac{4 \sqrt{2}}{9} + \frac{10 \sqrt{5}}{9}$$

      ___        ___
  4*\/ 2    10*\/ 5 
- ------- + --------
     9         9

$$- \frac{4 \sqrt{2}}{9} + \frac{10 \sqrt{5}}{9}$$

-4*sqrt(2)/9 + 10*sqrt(5)/9

Numerical answer [src]

1.85598061394506

1.85598061394506

The graph

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

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Integral of sqrt(5-3x) dx

Limits of integration:

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Piecewise:

The solution