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Identical expressions
//e^(three x)+3/2x
divide by divide by e to the power of (3x) plus 3 divide by 2x
divide by divide by e to the power of (three x) plus 3 divide by 2x
//e(3x)+3/2x
//e3x+3/2x
//e^3x+3/2x
divide by divide by e^(3x)+3 divide by 2x
//e^(3x)+3/2xdx
Similar expressions
//e^(3x)-3/2x

Integral of //e^(3x)+3/2x dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |  / 3*x   3*x\   
 |  |E    + ---| dx
 |  \        2 /   
 |                 
/                  
0

$$\int\limits_{0}^{1} \left(\frac{3 x}{2} + e^{3 x}\right)\, dx$$

Integral(E^(3*x) + 3*x/2, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{3 x}{2}\, dx = \frac{3 \int x\, dx}{2}$
  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: $\frac{3 x^{2}}{4}$
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{e^{u}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $\frac{e^{u}}{3}$
  Now substitute $u$ back in:
  
  $\frac{e^{3 x}}{3}$
The result is: $\frac{3 x^{2}}{4} + \frac{e^{3 x}}{3}$
Add the constant of integration:

$\frac{3 x^{2}}{4} + \frac{e^{3 x}}{3}+ \mathrm{constant}$

The answer is:

$\frac{3 x^{2}}{4} + \frac{e^{3 x}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                        3*x      2
 | / 3*x   3*x\          e      3*x 
 | |E    + ---| dx = C + ---- + ----
 | \        2 /           3      4  
 |                                  
/

$$\int \left(\frac{3 x}{2} + e^{3 x}\right)\, dx = C + \frac{3 x^{2}}{4} + \frac{e^{3 x}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$\frac{5}{12} + \frac{e^{3}}{3}$$

$$\frac{5}{12} + \frac{e^{3}}{3}$$

5/12 + exp(3)/3

Numerical answer [src]

7.11184564106256

7.11184564106256

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

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Similar expressions

Integral of //e^(3x)+3/2x dx

Limits of integration:

The graph:

Piecewise:

The solution