Integral of sqrt(7-4x) dx

The solution

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  1               
  /               
 |                
 |    _________   
 |  \/ 7 - 4*x  dx
 |                
/                 
0

$$\int\limits_{0}^{1} \sqrt{7 - 4 x}\, dx$$

Integral(sqrt(7 - 4*x), (x, 0, 1))

Detail solution

Let $u = 7 - 4 x$ .

Then let $du = - 4 dx$ and substitute $- \frac{du}{4}$ :

$\int \left(- \frac{\sqrt{u}}{4}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sqrt{u}\, du = - \frac{\int \sqrt{u}\, du}{4}$
  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: $- \frac{u^{\frac{3}{2}}}{6}$
Now substitute $u$ back in:

$- \frac{\left(7 - 4 x\right)^{\frac{3}{2}}}{6}$
Add the constant of integration:

$- \frac{\left(7 - 4 x\right)^{\frac{3}{2}}}{6}+ \mathrm{constant}$

The answer is:

$- \frac{\left(7 - 4 x\right)^{\frac{3}{2}}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                               3/2
 |   _________          (7 - 4*x)   
 | \/ 7 - 4*x  dx = C - ------------
 |                           6      
/

$$\int \sqrt{7 - 4 x}\, dx = C - \frac{\left(7 - 4 x\right)^{\frac{3}{2}}}{6}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    ___       ___
  \/ 3    7*\/ 7 
- ----- + -------
    2        6

$$- \frac{\sqrt{3}}{2} + \frac{7 \sqrt{7}}{6}$$

    ___       ___
  \/ 3    7*\/ 7 
- ----- + -------
    2        6

$$- \frac{\sqrt{3}}{2} + \frac{7 \sqrt{7}}{6}$$

-sqrt(3)/2 + 7*sqrt(7)/6

Numerical answer [src]

2.22068445912425

2.22068445912425

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

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Integral of sqrt(7-4x) dx

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The solution