Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of 1÷(x^2-1)
Integral of cos(x)^(-2)
Integral of 1/(x^2+2*x+5)
Integral of 1/(x^2+2*x)
Identical expressions
sqrt(two *x+ five)- three /x-exp(four *x)
square root of (2 multiply by x plus 5) minus 3 divide by x minus exponent of (4 multiply by x)
square root of (two multiply by x plus five) minus three divide by x minus exponent of (four multiply by x)
√(2*x+5)-3/x-exp(4*x)
sqrt(2x+5)-3/x-exp(4x)
sqrt2x+5-3/x-exp4x
sqrt(2*x+5)-3 divide by x-exp(4*x)
sqrt(2*x+5)-3/x-exp(4*x)dx
Similar expressions
sqrt(2*x+5)-3/x+exp(4*x)
sqrt(2*x+5)+3/x-exp(4*x)
sqrt(2*x-5)-3/x-exp(4*x)

Solving integrals/ sqrt(2*x+5)-3/x-exp(4*x)

Integral of sqrt(2x+5)-3/x-exp(4x) dx

The solution

You have entered [src]

 -1                            
  /                            
 |                             
 |  /  _________   3    4*x\   
 |  |\/ 2*x + 5  - - - e   | dx
 |  \              x       /   
 |                             
/                              
-2

$$\int\limits_{-2}^{-1} \left(\left(\sqrt{2 x + 5} - \frac{3}{x}\right) - e^{4 x}\right)\, dx$$

Integral(sqrt(2*x + 5) - 3/x - exp(4*x), (x, -2, -1))

Detail solution

Integrate term-by-term:
1. Integrate term-by-term:
  1. Let $u = 2 x + 5$ .
    
    Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{\sqrt{u}}{2}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sqrt{u}\, du = \frac{\int \sqrt{u}\, du}{2}$
      1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}$
      So, the result is: $\frac{u^{\frac{3}{2}}}{3}$
    Now substitute $u$ back in:
    
    $\frac{\left(2 x + 5\right)^{\frac{3}{2}}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{3}{x}\right)\, dx = - 3 \int \frac{1}{x}\, dx$
    1. The integral of $\frac{1}{x}$ is $\log{\left(x \right)}$ .
    So, the result is: $- 3 \log{\left(x \right)}$
  The result is: $\frac{\left(2 x + 5\right)^{\frac{3}{2}}}{3} - 3 \log{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- e^{4 x}\right)\, dx = - \int e^{4 x}\, dx$
  1. Let $u = 4 x$ .
    
    Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
    
    $\int \frac{e^{u}}{4}\, du$
    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: $\frac{e^{u}}{4}$
    Now substitute $u$ back in:
    
    $\frac{e^{4 x}}{4}$
  So, the result is: $- \frac{e^{4 x}}{4}$
The result is: $\frac{\left(2 x + 5\right)^{\frac{3}{2}}}{3} - \frac{e^{4 x}}{4} - 3 \log{\left(x \right)}$
Now simplify:

$\frac{\left(2 x + 5\right)^{\frac{3}{2}}}{3} - \frac{e^{4 x}}{4} - 3 \log{\left(x \right)}$
Add the constant of integration:

$\frac{\left(2 x + 5\right)^{\frac{3}{2}}}{3} - \frac{e^{4 x}}{4} - 3 \log{\left(x \right)}+ \mathrm{constant}$

The answer is:

$\frac{\left(2 x + 5\right)^{\frac{3}{2}}}{3} - \frac{e^{4 x}}{4} - 3 \log{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                
 |                                               4*x            3/2
 | /  _________   3    4*x\                     e      (2*x + 5)   
 | |\/ 2*x + 5  - - - e   | dx = C - 3*log(x) - ---- + ------------
 | \              x       /                      4          3      
 |                                                                 
/

$$\int \left(\left(\sqrt{2 x + 5} - \frac{3}{x}\right) - e^{4 x}\right)\, dx = C + \frac{\left(2 x + 5\right)^{\frac{3}{2}}}{3} - \frac{e^{4 x}}{4} - 3 \log{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                          -4    -8
  1     ___              e     e  
- - + \/ 3  + 3*log(2) - --- + ---
  3                       4     4

$$- \frac{1}{3} - \frac{1}{4 e^{4}} + \frac{1}{4 e^{8}} + \sqrt{3} + 3 \log{\left(2 \right)}$$

                          -4    -8
  1     ___              e     e  
- - + \/ 3  + 3*log(2) - --- + ---
  3                       4     4

$$- \frac{1}{3} - \frac{1}{4 e^{4}} + \frac{1}{4 e^{8}} + \sqrt{3} + 3 \log{\left(2 \right)}$$

-1/3 + sqrt(3) + 3*log(2) - exp(-4)/4 + exp(-8)/4

Numerical answer [src]

3.47366397185017

3.47366397185017

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of sqrt(2x+5)-3/x-exp(4x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of sqrt(2*x+5)-3/x-exp(4*x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of sqrt(2x+5)-3/x-exp(4x) dx