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Integral of d{x}:
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Integral of e^-t
Integral of sin(x)*dx/x
Integral of sin(2*x)/cos(2*x)
Derivative of:
x*sinx*cos2x
Identical expressions
x*sinx*cos2x
x multiply by sinus of x multiply by co sinus of e of 2x
xsinxcos2x
x*sinx*cos2xdx

Solving integrals/ x*sinx*cos2x

Integral of xsinxcos2x dx

The solution

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

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

Detail solution

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 $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)} \cos{\left(2 x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 1$ .

To find $v{\left(x \right)}$ :
1. Rewrite the integrand:
  
  $\sin{\left(x \right)} \cos{\left(2 x \right)} = 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)} - \sin{\left(x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = 2 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$
    1. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int \left(- u^{2}\right)\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int u^{2}\, du = - \int u^{2}\, du$
        
        The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int u^{2}\, du = \frac{u^{3}}{3}$
        
        So, the result is: $- \frac{u^{3}}{3}$
      Now substitute $u$ back in:
      
      $- \frac{\cos^{3}{\left(x \right)}}{3}$
    So, the result is: $- \frac{2 \cos^{3}{\left(x \right)}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin{\left(x \right)}\right)\, dx = - \int \sin{\left(x \right)}\, dx$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
    So, the result is: $\cos{\left(x \right)}$
  The result is: $- \frac{2 \cos^{3}{\left(x \right)}}{3} + \cos{\left(x \right)}$
Now evaluate the sub-integral.
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{2 \cos^{3}{\left(x \right)}}{3}\right)\, dx = - \frac{2 \int \cos^{3}{\left(x \right)}\, dx}{3}$
  1. Rewrite the integrand:
    
    $\cos^{3}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)}$
  2. There are multiple ways to do this integral.
    Method #1
    1. Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int \left(1 - u^{2}\right)\, du$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $- \frac{u^{3}}{3}$
      
      The result is: $- \frac{u^{3}}{3} + u$
      
      Now substitute $u$ back in:
      
      $- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$
    Method #2
    1. Rewrite the integrand:
      
      $\left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)} = - \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
      
      Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{3}{\left(x \right)}}{3}$
      
      So, the result is: $- \frac{\sin^{3}{\left(x \right)}}{3}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
      
      The result is: $- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$
    Method #3
    1. Rewrite the integrand:
      
      $\left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)} = - \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
      
      Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{3}{\left(x \right)}}{3}$
      
      So, the result is: $- \frac{\sin^{3}{\left(x \right)}}{3}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
      
      The result is: $- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$
  So, the result is: $\frac{2 \sin^{3}{\left(x \right)}}{9} - \frac{2 \sin{\left(x \right)}}{3}$
1. The integral of cosine is sine:
  
  $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
The result is: $\frac{2 \sin^{3}{\left(x \right)}}{9} + \frac{\sin{\left(x \right)}}{3}$
Now simplify:

$- \frac{2 x \cos^{3}{\left(x \right)}}{3} + x \cos{\left(x \right)} - \frac{2 \sin^{3}{\left(x \right)}}{9} - \frac{\sin{\left(x \right)}}{3}$
Add the constant of integration:

$- \frac{2 x \cos^{3}{\left(x \right)}}{3} + x \cos{\left(x \right)} - \frac{2 \sin^{3}{\left(x \right)}}{9} - \frac{\sin{\left(x \right)}}{3}+ \mathrm{constant}$

The answer is:

$- \frac{2 x \cos^{3}{\left(x \right)}}{3} + x \cos{\left(x \right)} - \frac{2 \sin^{3}{\left(x \right)}}{9} - \frac{\sin{\left(x \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                3                 /       3            \
 |                            2*sin (x)   sin(x)     |  2*cos (x)         |
 | x*sin(x)*cos(2*x) dx = C - --------- - ------ + x*|- --------- + cos(x)|
 |                                9         3        \      3             /
/

$$\int x \sin{\left(x \right)} \cos{\left(2 x \right)}\, dx = C + x \left(- \frac{2 \cos^{3}{\left(x \right)}}{3} + \cos{\left(x \right)}\right) - \frac{2 \sin^{3}{\left(x \right)}}{9} - \frac{\sin{\left(x \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  4*cos(1)*sin(2)   cos(1)*cos(2)   2*sin(1)*sin(2)   5*cos(2)*sin(1)
- --------------- + ------------- + --------------- + ---------------
         9                3                3                 9

$$- \frac{4 \sin{\left(2 \right)} \cos{\left(1 \right)}}{9} + \frac{5 \sin{\left(1 \right)} \cos{\left(2 \right)}}{9} + \frac{\cos{\left(1 \right)} \cos{\left(2 \right)}}{3} + \frac{2 \sin{\left(1 \right)} \sin{\left(2 \right)}}{3}$$

  4*cos(1)*sin(2)   cos(1)*cos(2)   2*sin(1)*sin(2)   5*cos(2)*sin(1)
- --------------- + ------------- + --------------- + ---------------
         9                3                3                 9

-4*cos(1)*sin(2)/9 + cos(1)*cos(2)/3 + 2*sin(1)*sin(2)/3 + 5*cos(2)*sin(1)/9

Numerical answer [src]

0.0222544104112996

0.0222544104112996

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of xsinxcos2x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3

Other calculators

How to use it?

Identical expressions

Integral of x*sinx*cos2x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3

Integral of xsinxcos2x dx