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Solving integrals/ -sqrt(4x+1)

Integral of -sqrt(4x+1) dx

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The solution

You have entered [src] 
  1                
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 |     _________   
 |  -\/ 4*x + 1  dx
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∫01(−4x+1) dx\int\limits_{0}^{1} \left(- \sqrt{4 x + 1}\right)\, dx 
Integral(-sqrt(4*x + 1), (x, 0, 1))
Detail solution

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

    ∫(−4x+1) dx=−∫4x+1 dx\int \left(- \sqrt{4 x + 1}\right)\, dx = - \int \sqrt{4 x + 1}\, dx

    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}

    So, the result is: −(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 \left(- \sqrt{4 x + 1}\right)\, dx = C - \frac{\left(4 x + 1\right)^{\frac{3}{2}}}{6}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-2.50.0
Plot the graph f(x)
The answer [src] 
        ___
1   5*\/ 5 
- - -------
6      6
16−556\frac{1}{6} - \frac{5 \sqrt{5}}{6} 
=
= 
        ___
1   5*\/ 5 
- - -------
6      6
16−556\frac{1}{6} - \frac{5 \sqrt{5}}{6} 
1/6 - 5*sqrt(5)/6
Numerical answer [src] 
-1.69672331458316
-1.69672331458316

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

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