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Integral of d{x}:
Integral of x^-1
Integral of xdx
Integral of x*x
Integral of cos^5(x)
Identical expressions
x^ seven *e^(x^ four)
x to the power of 7 multiply by e to the power of (x to the power of 4)
x to the power of seven multiply by e to the power of (x to the power of four)
x7*e(x4)
x7*ex4
x⁷*e^(x⁴)
x^7e^(x^4)
x7e(x4)
x7ex4
x^7e^x^4
x^7*e^(x^4)dx

Solving integrals/ x^7*e^(x^4)

Integral of x^7*e^(x^4) dx

The solution

You have entered [src]

  1            
  /            
 |             
 |      / 4\   
 |   7  \x /   
 |  x *E     dx
 |             
/              
0

$$\int\limits_{0}^{1} e^{x^{4}} x^{7}\, dx$$

Integral(x^7*E^(x^4), (x, 0, 1))

Detail solution

Let $u = x^{4}$ .

Then let $du = 4 x^{3} dx$ and substitute $\frac{du}{4}$ :

$\int \frac{u e^{u}}{4}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int u e^{u}\, du = \frac{\int u e^{u}\, du}{4}$
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
    
    Then $\operatorname{du}{\left(u \right)} = 1$ .
    
    To find $v{\left(u \right)}$ :
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    Now evaluate the sub-integral.
  2. The integral of the exponential function is itself.
    
    $\int e^{u}\, du = e^{u}$
  So, the result is: $\frac{u e^{u}}{4} - \frac{e^{u}}{4}$
Now substitute $u$ back in:

$\frac{x^{4} e^{x^{4}}}{4} - \frac{e^{x^{4}}}{4}$
Now simplify:

$\frac{\left(x^{4} - 1\right) e^{x^{4}}}{4}$
Add the constant of integration:

$\frac{\left(x^{4} - 1\right) e^{x^{4}}}{4}+ \mathrm{constant}$

The answer is:

$\frac{\left(x^{4} - 1\right) e^{x^{4}}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                  
 |                    / 4\       / 4\
 |     / 4\           \x /    4  \x /
 |  7  \x /          e       x *e    
 | x *E     dx = C - ----- + --------
 |                     4        4    
/

$$\int e^{x^{4}} x^{7}\, dx = C + \frac{x^{4} e^{x^{4}}}{4} - \frac{e^{x^{4}}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/4

$$\frac{1}{4}$$

1/4

$$\frac{1}{4}$$

1/4

Numerical answer [src]

0.25

0.25

The graph

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

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Identical expressions

Integral of x^7*e^(x^4) dx

Limits of integration:

The graph:

Piecewise:

The solution