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e^-x^4+5x^3dx
Solving integrals/ e^-x^4+5x^3dx

You entered:

e^-x^4+5x^3dx

What you mean?

e^((-x)^4) + 5*x^3*dx
 
e^(-x^4) + 5*x^3*dx
 
e^(-x^4) + 5*x^3*dx

Integral of e^-x^4+5x^3dx dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  1                   
  /                   
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 |  /   4         \   
 |  | -x       3  |   
 |  \e    + 5*x *1/ dx
 |                    
/                     
0
∫01(5x3⋅1+e−x4) dx\int\limits_{0}^{1} \left(5 x^{3} \cdot 1 + e^{- x^{4}}\right)\, dx 
Integral(E^(-x^4) + 5*x^3*1, (x, 0, 1))
Detail solution

  1. Integrate term-by-term:

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

      ∫5x3⋅1 dx=5∫x3 dx\int 5 x^{3} \cdot 1\, dx = 5 \int x^{3}\, dx

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫x3 dx=x44\int x^{3}\, dx = \frac{x^{4}}{4}

      So, the result is: 5x44\frac{5 x^{4}}{4}

    1. Don't know the steps in finding this integral.

      But the integral is

      Γ(14)γ(14,x4)16Γ(54)\frac{\Gamma\left(\frac{1}{4}\right) \gamma\left(\frac{1}{4}, x^{4}\right)}{16 \Gamma\left(\frac{5}{4}\right)}

    The result is: 5x44+Γ(14)γ(14,x4)16Γ(54)\frac{5 x^{4}}{4} + \frac{\Gamma\left(\frac{1}{4}\right) \gamma\left(\frac{1}{4}, x^{4}\right)}{16 \Gamma\left(\frac{5}{4}\right)}

  2. Now simplify:

    5x44+γ(14,x4)4\frac{5 x^{4}}{4} + \frac{\gamma\left(\frac{1}{4}, x^{4}\right)}{4}

  3. Add the constant of integration:

    5x44+γ(14,x4)4+constant\frac{5 x^{4}}{4} + \frac{\gamma\left(\frac{1}{4}, x^{4}\right)}{4}+ \mathrm{constant}

The answer is:

5x44+γ(14,x4)4+constant\frac{5 x^{4}}{4} + \frac{\gamma\left(\frac{1}{4}, x^{4}\right)}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                              
 |                                                               
 | /   4         \             4                        /      4\
 | | -x       3  |          5*x    Gamma(1/4)*lowergamma\1/4, x /
 | \e    + 5*x *1/ dx = C + ---- + ------------------------------
 |                           4             16*Gamma(5/4)         
/
∫(5x3⋅1+e−x4) dx=C+5x44+Γ(14)γ(14,x4)16Γ(54)\int \left(5 x^{3} \cdot 1 + e^{- x^{4}}\right)\, dx = C + \frac{5 x^{4}}{4} + \frac{\Gamma\left(\frac{1}{4}\right) \gamma\left(\frac{1}{4}, x^{4}\right)}{16 \Gamma\left(\frac{5}{4}\right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90010
Plot the graph f(x)
The answer [src] 
5   Gamma(1/4)*lowergamma(1/4, 1)
- + -----------------------------
4           16*Gamma(5/4)
Γ(14)γ(14,1)16Γ(54)+54\frac{\Gamma\left(\frac{1}{4}\right) \gamma\left(\frac{1}{4}, 1\right)}{16 \Gamma\left(\frac{5}{4}\right)} + \frac{5}{4} 
=
= 
5   Gamma(1/4)*lowergamma(1/4, 1)
- + -----------------------------
4           16*Gamma(5/4)
Γ(14)γ(14,1)16Γ(54)+54\frac{\Gamma\left(\frac{1}{4}\right) \gamma\left(\frac{1}{4}, 1\right)}{16 \Gamma\left(\frac{5}{4}\right)} + \frac{5}{4}
Упростить
Numerical answer [src] 
2.0948385947571
2.0948385947571
The graph
Integral of e^-x^4+5x^3dx dx

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

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