Mister Exam

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Integral of d{x}:
Integral of sinx
Integral of cosx^2
Integral of sinh(x)
Integral of xcos(nx)
Identical expressions
(sin(9x)-(x^(- three / four)))
( sinus of (9x) minus (x to the power of ( minus 3 divide by 4)))
( sinus of (9x) minus (x to the power of ( minus three divide by four)))
(sin(9x)-(x(-3/4)))
sin9x-x-3/4
sin9x-x^-3/4
(sin(9x)-(x^(-3 divide by 4)))
(sin(9x)-(x^(-3/4)))dx
Similar expressions
(sin(9x)-(x^(3/4)))
(sin(9x)+(x^(-3/4)))

Integral of (sin(9x)-(x^(-3/4))) dx

The solution

You have entered [src]

  1                     
  /                     
 |                      
 |  /            1  \   
 |  |sin(9*x) - ----| dx
 |  |            3/4|   
 |  \           x   /   
 |                      
/                       
0

$$\int\limits_{0}^{1} \left(\sin{\left(9 x \right)} - \frac{1}{x^{\frac{3}{4}}}\right)\, dx$$

Integral(sin(9*x) - 1/x^(3/4), (x, 0, 1))

Detail solution

Integrate term-by-term:
1. Let $u = 9 x$ .
  
  Then let $du = 9 dx$ and substitute $\frac{du}{9}$ :
  
  $\int \frac{\sin{\left(u \right)}}{9}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{9}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{9}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(9 x \right)}}{9}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{x^{\frac{3}{4}}}\right)\, dx = - \int \frac{1}{x^{\frac{3}{4}}}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \frac{1}{x^{\frac{3}{4}}}\, dx = 4 \sqrt[4]{x}$
  So, the result is: $- 4 \sqrt[4]{x}$
The result is: $- 4 \sqrt[4]{x} - \frac{\cos{\left(9 x \right)}}{9}$
Add the constant of integration:

$- 4 \sqrt[4]{x} - \frac{\cos{\left(9 x \right)}}{9}+ \mathrm{constant}$

The answer is:

$- 4 \sqrt[4]{x} - \frac{\cos{\left(9 x \right)}}{9}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                             
 |                                              
 | /            1  \            4 ___   cos(9*x)
 | |sin(9*x) - ----| dx = C - 4*\/ x  - --------
 | |            3/4|                       9    
 | \           x   /                            
 |                                              
/

$$\int \left(\sin{\left(9 x \right)} - \frac{1}{x^{\frac{3}{4}}}\right)\, dx = C - 4 \sqrt[4]{x} - \frac{\cos{\left(9 x \right)}}{9}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  35   cos(9)
- -- - ------
  9      9

$$- \frac{35}{9} - \frac{\cos{\left(9 \right)}}{9}$$

  35   cos(9)
- -- - ------
  9      9

$$- \frac{35}{9} - \frac{\cos{\left(9 \right)}}{9}$$

-35/9 - cos(9)/9

Numerical answer [src]

-3.78758696103453

-3.78758696103453

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

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Similar expressions

Integral of (sin(9x)-(x^(-3/4))) dx

Limits of integration:

The graph:

Piecewise:

The solution