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Integral of exp(((x+1)^4)/8)*y dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  t               
  /               
 |                
 |          4     
 |   (x + 1)      
 |   --------     
 |      8         
 |  e        *y dx
 |                
/                 
0                 
$$\int\limits_{0}^{t} y e^{\frac{\left(x + 1\right)^{4}}{8}}\, dx$$
Integral(exp((x + 1)^4/8)*y, (x, 0, t))
Detail solution
  1. The integral of a constant times a function is the constant times the integral of the function:

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

      But the integral is

    So, the result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                              
 |                              -pi*I                                            
 |         4                    ------                      /            4  pi*I\
 |  (x + 1)                3/4    4                         |     (1 + x) *e    |
 |  --------            y*2   *e      *Gamma(1/4)*lowergamma|1/4, --------------|
 |     8                                                    \           8       /
 | e        *y dx = C + ---------------------------------------------------------
 |                                            16*Gamma(5/4)                      
/                                                                                
$$\int y e^{\frac{\left(x + 1\right)^{4}}{8}}\, dx = C + \frac{2^{\frac{3}{4}} y e^{- \frac{i \pi}{4}} \Gamma\left(\frac{1}{4}\right) \gamma\left(\frac{1}{4}, \frac{\left(x + 1\right)^{4} e^{i \pi}}{8}\right)}{16 \Gamma\left(\frac{5}{4}\right)}$$
The answer [src]
          -pi*I                                              -pi*I                                            
          ------                      /      pi*I\           ------                      /            4  pi*I\
     3/4    4                         |     e    |      3/4    4                         |     (1 + t) *e    |
  y*2   *e      *Gamma(1/4)*lowergamma|1/4, -----|   y*2   *e      *Gamma(1/4)*lowergamma|1/4, --------------|
                                      \       8  /                                       \           8       /
- ------------------------------------------------ + ---------------------------------------------------------
                   16*Gamma(5/4)                                           16*Gamma(5/4)                      
$$\frac{2^{\frac{3}{4}} y e^{- \frac{i \pi}{4}} \Gamma\left(\frac{1}{4}\right) \gamma\left(\frac{1}{4}, \frac{\left(t + 1\right)^{4} e^{i \pi}}{8}\right)}{16 \Gamma\left(\frac{5}{4}\right)} - \frac{2^{\frac{3}{4}} y e^{- \frac{i \pi}{4}} \Gamma\left(\frac{1}{4}\right) \gamma\left(\frac{1}{4}, \frac{e^{i \pi}}{8}\right)}{16 \Gamma\left(\frac{5}{4}\right)}$$
=
=
          -pi*I                                              -pi*I                                            
          ------                      /      pi*I\           ------                      /            4  pi*I\
     3/4    4                         |     e    |      3/4    4                         |     (1 + t) *e    |
  y*2   *e      *Gamma(1/4)*lowergamma|1/4, -----|   y*2   *e      *Gamma(1/4)*lowergamma|1/4, --------------|
                                      \       8  /                                       \           8       /
- ------------------------------------------------ + ---------------------------------------------------------
                   16*Gamma(5/4)                                           16*Gamma(5/4)                      
$$\frac{2^{\frac{3}{4}} y e^{- \frac{i \pi}{4}} \Gamma\left(\frac{1}{4}\right) \gamma\left(\frac{1}{4}, \frac{\left(t + 1\right)^{4} e^{i \pi}}{8}\right)}{16 \Gamma\left(\frac{5}{4}\right)} - \frac{2^{\frac{3}{4}} y e^{- \frac{i \pi}{4}} \Gamma\left(\frac{1}{4}\right) \gamma\left(\frac{1}{4}, \frac{e^{i \pi}}{8}\right)}{16 \Gamma\left(\frac{5}{4}\right)}$$
-y*2^(3/4)*exp(-pi*i/4)*gamma(1/4)*lowergamma(1/4, exp_polar(pi*i)/8)/(16*gamma(5/4)) + y*2^(3/4)*exp(-pi*i/4)*gamma(1/4)*lowergamma(1/4, (1 + t)^4*exp_polar(pi*i)/8)/(16*gamma(5/4))

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