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Identical expressions
exp(((x+ one)^ four)/ eight)*y
exponent of (((x plus 1) to the power of 4) divide by 8) multiply by y
exponent of (((x plus one) to the power of four) divide by eight) multiply by y
exp(((x+1)4)/8)*y
expx+14/8*y
exp(((x+1)⁴)/8)*y
exp(((x+1)^4)/8)y
exp(((x+1)4)/8)y
expx+14/8y
expx+1^4/8y
exp(((x+1)^4) divide by 8)*y
exp(((x+1)^4)/8)*ydx
Similar expressions
exp(((x-1)^4)/8)*y

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

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

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

$\int y e^{\frac{\left(x + 1\right)^{4}}{8}}\, dx = y \int e^{\frac{\left(x + 1\right)^{4}}{8}}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{2^{\frac{3}{4}} e^{- \frac{i \pi}{4}} \Gamma\left(\frac{1}{4}\right) \gamma\left(\frac{1}{4}, \frac{\left(x + 1\right)^{4} e^{i \pi}}{8}\right)}{16 \Gamma\left(\frac{5}{4}\right)}$
So, the result is: $\frac{2^{\frac{3}{4}} y e^{- \frac{i \pi}{4}} \Gamma\left(\frac{1}{4}\right) \gamma\left(\frac{1}{4}, \frac{\left(x + 1\right)^{4} e^{i \pi}}{8}\right)}{16 \Gamma\left(\frac{5}{4}\right)}$
Now simplify:

$- \frac{\left(-2\right)^{\frac{3}{4}} y \gamma\left(\frac{1}{4}, \frac{\left(x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1\right) e^{i \pi}}{8}\right)}{4}$
Add the constant of integration:

$- \frac{\left(-2\right)^{\frac{3}{4}} y \gamma\left(\frac{1}{4}, \frac{\left(x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1\right) e^{i \pi}}{8}\right)}{4}+ \mathrm{constant}$

The answer is:

$- \frac{\left(-2\right)^{\frac{3}{4}} y \gamma\left(\frac{1}{4}, \frac{\left(x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1\right) e^{i \pi}}{8}\right)}{4}+ \mathrm{constant}$

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)                      
/

$$\int y e^{\frac{\left(x + 1\right)^{4}}{8}}\, dx = C + \frac{2^{\frac{3}{4}} y e^{- \frac{i \pi}{4}} \Gamma\left(\frac{1}{4}\right) \gamma\left(\frac{1}{4}, \frac{\left(x + 1\right)^{4} e^{i \pi}}{8}\right)}{16 \Gamma\left(\frac{5}{4}\right)}$$

Simplify

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)

-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.

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

Limits of integration:

The graph:

Piecewise:

The solution