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Identical expressions
(15x+ four)*sin5x
(15x plus 4) multiply by sinus of 5x
(15x plus four) multiply by sinus of 5x
(15x+4)sin5x
15x+4sin5x
(15x+4)*sin5xdx
Similar expressions
(15x-4)*sin5x

Integral of (15x+4)*sin5x dx

The solution

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 pi                       
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 3                        
  /                       
 |                        
 |  (15*x + 4)*sin(5*x) dx
 |                        
/                         
0

$$\int\limits_{0}^{\frac{\pi}{3}} \left(15 x + 4\right) \sin{\left(5 x \right)}\, dx$$

Integral((15*x + 4)*sin(5*x), (x, 0, pi/3))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 5 x$ .
  
  Then let $du = 5 dx$ and substitute $du$ :
  
  $\int \left(\frac{3 u \sin{\left(u \right)}}{5} + \frac{4 \sin{\left(u \right)}}{5}\right)\, du$
  1. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{3 u \sin{\left(u \right)}}{5}\, du = \frac{3 \int u \sin{\left(u \right)}\, du}{5}$
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = \sin{\left(u \right)}$ .
      
      Then $\operatorname{du}{\left(u \right)} = 1$ .
      
      To find $v{\left(u \right)}$ :
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      Now evaluate the sub-integral.
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \cos{\left(u \right)}\right)\, du = - \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $- \sin{\left(u \right)}$
      
      So, the result is: $- \frac{3 u \cos{\left(u \right)}}{5} + \frac{3 \sin{\left(u \right)}}{5}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{4 \sin{\left(u \right)}}{5}\, du = \frac{4 \int \sin{\left(u \right)}\, du}{5}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{4 \cos{\left(u \right)}}{5}$
    The result is: $- \frac{3 u \cos{\left(u \right)}}{5} + \frac{3 \sin{\left(u \right)}}{5} - \frac{4 \cos{\left(u \right)}}{5}$
  Now substitute $u$ back in:
  
  $- 3 x \cos{\left(5 x \right)} + \frac{3 \sin{\left(5 x \right)}}{5} - \frac{4 \cos{\left(5 x \right)}}{5}$
Method #2
1. Rewrite the integrand:
  
  $\left(15 x + 4\right) \sin{\left(5 x \right)} = 15 x \sin{\left(5 x \right)} + 4 \sin{\left(5 x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 15 x \sin{\left(5 x \right)}\, dx = 15 \int x \sin{\left(5 x \right)}\, dx$
    1. 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 $\operatorname{dv}{\left(x \right)} = \sin{\left(5 x \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{\sin{\left(u \right)}}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{5}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos{\left(5 x \right)}}{5}$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{\cos{\left(5 x \right)}}{5}\right)\, dx = - \frac{\int \cos{\left(5 x \right)}\, dx}{5}$
      
      Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{\cos{\left(u \right)}}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{5}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(5 x \right)}}{5}$
      
      So, the result is: $- \frac{\sin{\left(5 x \right)}}{25}$
    So, the result is: $- 3 x \cos{\left(5 x \right)} + \frac{3 \sin{\left(5 x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 4 \sin{\left(5 x \right)}\, dx = 4 \int \sin{\left(5 x \right)}\, dx$
    1. Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{\sin{\left(u \right)}}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{5}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos{\left(5 x \right)}}{5}$
    So, the result is: $- \frac{4 \cos{\left(5 x \right)}}{5}$
  The result is: $- 3 x \cos{\left(5 x \right)} + \frac{3 \sin{\left(5 x \right)}}{5} - \frac{4 \cos{\left(5 x \right)}}{5}$
Method #3
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = 15 x + 4$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(5 x \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = 15$ .
  
  To find $v{\left(x \right)}$ :
  1. Let $u = 5 x$ .
    
    Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
    
    $\int \frac{\sin{\left(u \right)}}{5}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{5}$
    Now substitute $u$ back in:
    
    $- \frac{\cos{\left(5 x \right)}}{5}$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 3 \cos{\left(5 x \right)}\right)\, dx = - 3 \int \cos{\left(5 x \right)}\, dx$
  1. Let $u = 5 x$ .
    
    Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
    
    $\int \frac{\cos{\left(u \right)}}{5}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{5}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{5}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(5 x \right)}}{5}$
  So, the result is: $- \frac{3 \sin{\left(5 x \right)}}{5}$
Method #4
1. Rewrite the integrand:
  
  $\left(15 x + 4\right) \sin{\left(5 x \right)} = 15 x \sin{\left(5 x \right)} + 4 \sin{\left(5 x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 15 x \sin{\left(5 x \right)}\, dx = 15 \int x \sin{\left(5 x \right)}\, dx$
    1. 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 $\operatorname{dv}{\left(x \right)} = \sin{\left(5 x \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{\sin{\left(u \right)}}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{5}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos{\left(5 x \right)}}{5}$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{\cos{\left(5 x \right)}}{5}\right)\, dx = - \frac{\int \cos{\left(5 x \right)}\, dx}{5}$
      
      Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{\cos{\left(u \right)}}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{5}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(5 x \right)}}{5}$
      
      So, the result is: $- \frac{\sin{\left(5 x \right)}}{25}$
    So, the result is: $- 3 x \cos{\left(5 x \right)} + \frac{3 \sin{\left(5 x \right)}}{5}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 4 \sin{\left(5 x \right)}\, dx = 4 \int \sin{\left(5 x \right)}\, dx$
    1. Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{\sin{\left(u \right)}}{5}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{5}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos{\left(5 x \right)}}{5}$
    So, the result is: $- \frac{4 \cos{\left(5 x \right)}}{5}$
  The result is: $- 3 x \cos{\left(5 x \right)} + \frac{3 \sin{\left(5 x \right)}}{5} - \frac{4 \cos{\left(5 x \right)}}{5}$
Add the constant of integration:

$- 3 x \cos{\left(5 x \right)} + \frac{3 \sin{\left(5 x \right)}}{5} - \frac{4 \cos{\left(5 x \right)}}{5}+ \mathrm{constant}$

The answer is:

$- 3 x \cos{\left(5 x \right)} + \frac{3 \sin{\left(5 x \right)}}{5} - \frac{4 \cos{\left(5 x \right)}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                   
 |                              4*cos(5*x)   3*sin(5*x)               
 | (15*x + 4)*sin(5*x) dx = C - ---------- + ---------- - 3*x*cos(5*x)
 |                                  5            5                    
/

$$\int \left(15 x + 4\right) \sin{\left(5 x \right)}\, dx = C - 3 x \cos{\left(5 x \right)} + \frac{3 \sin{\left(5 x \right)}}{5} - \frac{4 \cos{\left(5 x \right)}}{5}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

        ___     
2   3*\/ 3    pi
- - ------- - --
5      10     2

$$- \frac{\pi}{2} - \frac{3 \sqrt{3}}{10} + \frac{2}{5}$$

        ___     
2   3*\/ 3    pi
- - ------- - --
5      10     2

$$- \frac{\pi}{2} - \frac{3 \sqrt{3}}{10} + \frac{2}{5}$$

2/5 - 3*sqrt(3)/10 - pi/2

Numerical answer [src]

-1.69041156906556

-1.69041156906556

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

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Identical expressions

Similar expressions

Integral of (15x+4)*sin5x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3

Method #4