Integral of sin(x)*cos(2x) dx

The solution

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

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

Detail solution

Rewrite the integrand:

$\sin{\left(x \right)} \cos{\left(2 x \right)} = 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)} - \sin{\left(x \right)}$
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:
      
      $\int u^{2}\, du = - \int u^{2}\, du$
      1. 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 simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1   cos(1)*cos(2)   2*sin(1)*sin(2)
- - + ------------- + ---------------
  3         3                3

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

  1   cos(1)*cos(2)   2*sin(1)*sin(2)
- - + ------------- + ---------------
  3         3                3

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

-1/3 + cos(1)*cos(2)/3 + 2*sin(1)*sin(2)/3

Numerical answer [src]

0.101816569034144

0.101816569034144

The graph

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

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Integral of sin(x)*cos(2x) dx

Limits of integration:

The graph:

Piecewise:

The solution