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Derivative of:
x*sin(2*x)
Limit of the function:
x*sin(2*x)
Identical expressions
x*sin(two *x)
x multiply by sinus of (2 multiply by x)
x multiply by sinus of (two multiply by x)
xsin(2x)
xsin2x
x*sin(2*x)dx
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Solving integrals/ x*sin(2*x)

Integral of xsin(2x) dx

The solution

You have entered [src]

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$$\int\limits_{0}^{1} x \sin{\left(2 x \right)}\, dx$$

Integral(x*sin(2*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
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 $\operatorname{dv}{\left(x \right)} = \sin{\left(2 x \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = 1$ .
  
  To find $v{\left(x \right)}$ :
  1. There are multiple ways to do this integral.
    Method #1
    
    Let $u = 2 x$ .
    
    Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{\sin{\left(u \right)}}{4}\, du$
    
    The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sin{\left(u \right)}}{2}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}$
    
    The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    
    So, the result is: $- \frac{\cos{\left(u \right)}}{2}$
    
    Now substitute $u$ back in:
    
    $- \frac{\cos{\left(2 x \right)}}{2}$
    Method #2
    
    The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 \sin{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int \sin{\left(x \right)} \cos{\left(x \right)}\, dx$
    
    Let $u = \cos{\left(x \right)}$ .
    
    Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
    
    $\int u\, du$
    
    The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- u\right)\, du = - \int u\, du$
    
    The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int u\, du = \frac{u^{2}}{2}$
    
    So, the result is: $- \frac{u^{2}}{2}$
    
    Now substitute $u$ back in:
    
    $- \frac{\cos^{2}{\left(x \right)}}{2}$
    
    So, the result is: $- \cos^{2}{\left(x \right)}$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
  1. Let $u = 2 x$ .
    
    Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{\cos{\left(u \right)}}{4}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{2}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(2 x \right)}}{2}$
  So, the result is: $- \frac{\sin{\left(2 x \right)}}{4}$
Method #2
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 2 x \sin{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int x \sin{\left(x \right)} \cos{\left(x \right)}\, dx$
  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 $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)} \cos{\left(x \right)}$ .
    
    Then $\operatorname{du}{\left(x \right)} = 1$ .
    
    To find $v{\left(x \right)}$ :
    1. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int u\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u\right)\, du = - \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{u^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{2}{\left(x \right)}}{2}$
    Now evaluate the sub-integral.
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{\cos^{2}{\left(x \right)}}{2}\right)\, dx = - \frac{\int \cos^{2}{\left(x \right)}\, dx}{2}$
    1. Rewrite the integrand:
      
      $\cos^{2}{\left(x \right)} = \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(2 x \right)}}{2}\, dx = \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
      
      Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{4}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
      
      So, the result is: $\frac{\sin{\left(2 x \right)}}{4}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
      
      The result is: $\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}$
    So, the result is: $- \frac{x}{4} - \frac{\sin{\left(2 x \right)}}{8}$
  So, the result is: $- x \cos^{2}{\left(x \right)} + \frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}$
Add the constant of integration:

$- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}$

The answer is:

$- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                         
 |                     sin(2*x)   x*cos(2*x)
 | x*sin(2*x) dx = C + -------- - ----------
 |                        4           2     
/

$${{\sin \left(2\,x\right)-2\,x\,\cos \left(2\,x\right)}\over{4}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  cos(2)   sin(2)
- ------ + ------
    2        4

$${{\sin 2-2\,\cos 2}\over{4}}$$

  cos(2)   sin(2)
- ------ + ------
    2        4

$$- \frac{\cos{\left(2 \right)}}{2} + \frac{\sin{\left(2 \right)}}{4}$$

Simplify

Numerical answer [src]

0.435397774979992

0.435397774979992

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of xsin(2x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #1

Method #2

Method #2

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of x*sin(2*x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #1

Method #2

Method #2

Integral of xsin(2x) dx