Integral of sin2xcosxdx dx

You have entered [src]

  0                     
  /                     
 |                      
 |  sin(2*x)*cos(x)*1 dx
 |                      
/                       
0

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
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 $- du$ :
    
    $\int u^{2}\, du$
    1. 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: $- \frac{u^{3}}{3}$
    Now substitute $u$ back in:
    
    $- \frac{\cos^{3}{\left(x \right)}}{3}$
  So, the result is: $- \frac{2 \cos^{3}{\left(x \right)}}{3}$
Method #2
1. Rewrite the integrand:
  
  $\sin{\left(2 x \right)} \cos{\left(x \right)} 1 = 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = 2 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$
  1. Let $u = \cos{\left(x \right)}$ .
    
    Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
    
    $\int u^{2}\, du$
    1. 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: $- \frac{u^{3}}{3}$
    Now substitute $u$ back in:
    
    $- \frac{\cos^{3}{\left(x \right)}}{3}$
  So, the result is: $- \frac{2 \cos^{3}{\left(x \right)}}{3}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                3   
 |                            2*cos (x)
 | sin(2*x)*cos(x)*1 dx = C - ---------
 |                                3    
/

$$-{{\cos \left(3\,x\right)}\over{6}}-{{\cos x}\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

=

$$0$$

Simplify

Numerical answer [src]

0.0

0.0

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of sin2xcosxdx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2