Mister Exam

Other calculators

Solving integrals/ sqrt(4*x+1)

Integral of sqrt(4*x+1) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  2               
  /               
 |                
 |    _________   
 |  \/ 4*x + 1  dx
 |                
/                 
1
∫124x+1 dx\int\limits_{1}^{2} \sqrt{4 x + 1}\, dx 
Integral(sqrt(4*x + 1), (x, 1, 2))
Detail solution

  1. Let u=4x+1u = 4 x + 1.

    Then let du=4dxdu = 4 dx and substitute du4\frac{du}{4}:

    ∫u4 du\int \frac{\sqrt{u}}{4}\, du

    1. The integral of a constant times a function is the constant times the integral of the function:

      ∫u du=∫u du4\int \sqrt{u}\, du = \frac{\int \sqrt{u}\, du}{4}

      1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫u du=2u323\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}

      So, the result is: u326\frac{u^{\frac{3}{2}}}{6}

    Now substitute uu back in:

    (4x+1)326\frac{\left(4 x + 1\right)^{\frac{3}{2}}}{6}

  2. Now simplify:

    (4x+1)326\frac{\left(4 x + 1\right)^{\frac{3}{2}}}{6}

  3. Add the constant of integration:

    (4x+1)326+constant\frac{\left(4 x + 1\right)^{\frac{3}{2}}}{6}+ \mathrm{constant}

The answer is:

(4x+1)326+constant\frac{\left(4 x + 1\right)^{\frac{3}{2}}}{6}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                 
 |                               3/2
 |   _________          (4*x + 1)   
 | \/ 4*x + 1  dx = C + ------------
 |                           6      
/
∫4x+1 dx=C+(4x+1)326\int \sqrt{4 x + 1}\, dx = C + \frac{\left(4 x + 1\right)^{\frac{3}{2}}}{6}
Simplify
The graph
1.002.001.101.201.301.401.501.601.701.801.900.05.0
Plot the graph f(x)
The answer [src] 
        ___
9   5*\/ 5 
- - -------
2      6
92−556\frac{9}{2} - \frac{5 \sqrt{5}}{6} 
=
= 
        ___
9   5*\/ 5 
- - -------
2      6
92−556\frac{9}{2} - \frac{5 \sqrt{5}}{6} 
9/2 - 5*sqrt(5)/6
Numerical answer [src] 
2.63661001875018
2.63661001875018

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

    © Mister Exam – Calculator