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Identical expressions
-x^ two +7x+12e^x
minus x squared plus 7x plus 12e to the power of x
minus x to the power of two plus 7x plus 12e to the power of x
-x2+7x+12ex
-x²+7x+12e^x
-x to the power of 2+7x+12e to the power of x
-x^2+7x+12e^xdx
Similar expressions
-x^2-7x+12e^x
x^2+7x+12e^x
-x^2+7x-12e^x

Integral of -x^2+7x+12e^x dx

The solution

You have entered [src]

 231                       
 ---                       
  25                       
  /                        
 |                         
 |  /   2             x\   
 |  \- x  + 7*x + 12*E / dx
 |                         
/                          
-1

$$\int\limits_{-1}^{\frac{231}{25}} \left(12 e^{x} + \left(- x^{2} + 7 x\right)\right)\, dx$$

Integral(-x^2 + 7*x + 12*E^x, (x, -1, 231/25))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 12 e^{x}\, dx = 12 \int e^{x}\, dx$
  1. The integral of the exponential function is itself.
    
    $\int e^{x}\, dx = e^{x}$
  So, the result is: $12 e^{x}$
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- x^{2}\right)\, dx = - \int x^{2}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    So, the result is: $- \frac{x^{3}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 7 x\, dx = 7 \int x\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
    So, the result is: $\frac{7 x^{2}}{2}$
  The result is: $- \frac{x^{3}}{3} + \frac{7 x^{2}}{2}$
The result is: $- \frac{x^{3}}{3} + \frac{7 x^{2}}{2} + 12 e^{x}$
Add the constant of integration:

$- \frac{x^{3}}{3} + \frac{7 x^{2}}{2} + 12 e^{x}+ \mathrm{constant}$

The answer is:

$- \frac{x^{3}}{3} + \frac{7 x^{2}}{2} + 12 e^{x}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                               
 |                                        3      2
 | /   2             x\              x   x    7*x 
 | \- x  + 7*x + 12*E / dx = C + 12*e  - -- + ----
 |                                       3     2  
/

$$\int \left(12 e^{x} + \left(- x^{2} + 7 x\right)\right)\, dx = C - \frac{x^{3}}{3} + \frac{7 x^{2}}{2} + 12 e^{x}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                       231
                       ---
1501184       -1        25
------- - 12*e   + 12*e   
 46875

$$- \frac{12}{e} + \frac{1501184}{46875} + 12 e^{\frac{231}{25}}$$

                       231
                       ---
1501184       -1        25
------- - 12*e   + 12*e   
 46875

$$- \frac{12}{e} + \frac{1501184}{46875} + 12 e^{\frac{231}{25}}$$

1501184/46875 - 12*exp(-1) + 12*exp(231/25)

Numerical answer [src]

123640.073400331

123640.073400331

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

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Integral of -x^2+7x+12e^x dx

Limits of integration:

The graph:

Piecewise:

The solution