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Identical expressions
(three - four x)sinx/4
(3 minus 4x) sinus of x divide by 4
(three minus four x) sinus of x divide by 4
3-4xsinx/4
(3-4x)sinx divide by 4
(3-4x)sinx/4dx
Similar expressions
(3+4x)sinx/4

Integral of (3-4x)sinx/4 dx

The solution

You have entered [src]

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

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

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

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{\left(3 - 4 x\right) \sin{\left(x \right)}}{4}\, dx = \frac{\int \left(3 - 4 x\right) \sin{\left(x \right)}\, dx}{4}$
1. There are multiple ways to do this integral.
  Method #1
  1. Rewrite the integrand:
    
    $\left(3 - 4 x\right) \sin{\left(x \right)} = - 4 x \sin{\left(x \right)} + 3 \sin{\left(x \right)}$
  2. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 4 x \sin{\left(x \right)}\right)\, dx = - 4 \int x \sin{\left(x \right)}\, dx$
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
      
      Now evaluate the sub-integral.
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \cos{\left(x \right)}\right)\, dx = - \int \cos{\left(x \right)}\, dx$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
      
      So, the result is: $- \sin{\left(x \right)}$
      
      So, the result is: $4 x \cos{\left(x \right)} - 4 \sin{\left(x \right)}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 3 \sin{\left(x \right)}\, dx = 3 \int \sin{\left(x \right)}\, dx$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
      
      So, the result is: $- 3 \cos{\left(x \right)}$
    The result is: $4 x \cos{\left(x \right)} - 4 \sin{\left(x \right)} - 3 \cos{\left(x \right)}$
  Method #2
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(x \right)} = 3 - 4 x$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .
    
    Then $\operatorname{du}{\left(x \right)} = -4$ .
    
    To find $v{\left(x \right)}$ :
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
    Now evaluate the sub-integral.
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 4 \cos{\left(x \right)}\, dx = 4 \int \cos{\left(x \right)}\, dx$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
    So, the result is: $4 \sin{\left(x \right)}$
So, the result is: $x \cos{\left(x \right)} - \sin{\left(x \right)} - \frac{3 \cos{\left(x \right)}}{4}$
Add the constant of integration:

$x \cos{\left(x \right)} - \sin{\left(x \right)} - \frac{3 \cos{\left(x \right)}}{4}+ \mathrm{constant}$

The answer is:

$x \cos{\left(x \right)} - \sin{\left(x \right)} - \frac{3 \cos{\left(x \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                      
 |                                                       
 | (3 - 4*x)*sin(x)                   3*cos(x)           
 | ---------------- dx = C - sin(x) - -------- + x*cos(x)
 |        4                              4               
 |                                                       
/

$$\int \frac{\left(3 - 4 x\right) \sin{\left(x \right)}}{4}\, dx = C + x \cos{\left(x \right)} - \sin{\left(x \right)} - \frac{3 \cos{\left(x \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

3            cos(1)
- - sin(1) + ------
4              4

$$- \sin{\left(1 \right)} + \frac{\cos{\left(1 \right)}}{4} + \frac{3}{4}$$

3            cos(1)
- - sin(1) + ------
4              4

$$- \sin{\left(1 \right)} + \frac{\cos{\left(1 \right)}}{4} + \frac{3}{4}$$

3/4 - sin(1) + cos(1)/4

Numerical answer [src]

0.0436045916591384

0.0436045916591384

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

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Identical expressions

Similar expressions

Integral of (3-4x)sinx/4 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2