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√(3x²-2x+4)
Solving integrals/ √(3x²-2x+4)

Integral of √(3x²-2x+4) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src] 
  1                       
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 |  \/  3*x  - 2*x + 4  dx
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0
∫013x2−2x+4 dx\int\limits_{0}^{1} \sqrt{3 x^{2} - 2 x + 4}\, dx 
Integral(sqrt(3*x^2 - 2*x + 4), (x, 0, 1))
Detail solution

    SqrtQuadraticRule(a=4, b=-2, c=3, context=sqrt(3*x**2 - 2*x + 4), symbol=x)

  1. Now simplify:

    (3x−1)3x2−2x+46+113asinh⁡(11⋅(3x−1)11)18\frac{\left(3 x - 1\right) \sqrt{3 x^{2} - 2 x + 4}}{6} + \frac{11 \sqrt{3} \operatorname{asinh}{\left(\frac{\sqrt{11} \cdot \left(3 x - 1\right)}{11} \right)}}{18}

  2. Add the constant of integration:

    (3x−1)3x2−2x+46+113asinh⁡(11⋅(3x−1)11)18+constant\frac{\left(3 x - 1\right) \sqrt{3 x^{2} - 2 x + 4}}{6} + \frac{11 \sqrt{3} \operatorname{asinh}{\left(\frac{\sqrt{11} \cdot \left(3 x - 1\right)}{11} \right)}}{18}+ \mathrm{constant}

The answer is:

(3x−1)3x2−2x+46+113asinh⁡(11⋅(3x−1)11)18+constant\frac{\left(3 x - 1\right) \sqrt{3 x^{2} - 2 x + 4}}{6} + \frac{11 \sqrt{3} \operatorname{asinh}{\left(\frac{\sqrt{11} \cdot \left(3 x - 1\right)}{11} \right)}}{18}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                           /    ____           \
 |                                                                   ___      |3*\/ 11 *(-1/3 + x)|
 |    ________________             ________________             11*\/ 3 *asinh|-------------------|
 |   /    2                       /              2  /  1   x\                 \         11        /
 | \/  3*x  - 2*x + 4  dx = C + \/  4 - 2*x + 3*x  *|- - + -| + -----------------------------------
 |                                                  \  6   2/                    18                
/
11 asinh  (6 x−22 11)2 332+x 3 x2−2 x+42−3 x2−2 x+46{{11\,{\rm asinh}\; \left({{6\,x-2}\over{2\,\sqrt{11}}}\right) }\over{2\,3^{{{3}\over{2}}}}}+{{x\,\sqrt{3\,x^2-2\,x+4}}\over{2}}-{{ \sqrt{3\,x^2-2\,x+4}}\over{6}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.905-5
Plot the graph f(x)
The answer [src] 
                          /  ____\                 /    ____\
                 ___      |\/ 11 |        ___      |2*\/ 11 |
      ___   11*\/ 3 *asinh|------|   11*\/ 3 *asinh|--------|
1   \/ 5                  \  11  /                 \   11   /
- + ----- + ---------------------- + ------------------------
3     3               18                        18
11 3 asinh  (211)+6 518+11 3 asinh  (111)+618{{11\,\sqrt{3}\,{\rm asinh}\; \left({{2}\over{\sqrt{11}}}\right)+6 \,\sqrt{5}}\over{18}}+{{11\,\sqrt{3}\,{\rm asinh}\; \left({{1}\over{ \sqrt{11}}}\right)+6}\over{18}} 
=
= 
                          /  ____\                 /    ____\
                 ___      |\/ 11 |        ___      |2*\/ 11 |
      ___   11*\/ 3 *asinh|------|   11*\/ 3 *asinh|--------|
1   \/ 5                  \  11  /                 \   11   /
- + ----- + ---------------------- + ------------------------
3     3               18                        18
113asinh⁡(1111)18+13+113asinh⁡(21111)18+53\frac{11 \sqrt{3} \operatorname{asinh}{\left(\frac{\sqrt{11}}{11} \right)}}{18} + \frac{1}{3} + \frac{11 \sqrt{3} \operatorname{asinh}{\left(\frac{2 \sqrt{11}}{11} \right)}}{18} + \frac{\sqrt{5}}{3}
Simplify
Numerical answer [src] 
1.99801282141864
1.99801282141864
The graph
Integral of √(3x²-2x+4) dx

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

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