Integral of sin2t*cost dx

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 pi                   
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 |  sin(2*t)*cos(t) dt
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0

$$\int\limits_{0}^{\pi} \sin{\left(2 t \right)} \cos{\left(t \right)}\, dt$$

Integral(sin(2*t)*cos(t), (t, 0, pi))

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:
  
  $\int 2 \sin{\left(t \right)} \cos^{2}{\left(t \right)}\, dt = 2 \int \sin{\left(t \right)} \cos^{2}{\left(t \right)}\, dt$
  1. Let $u = \cos{\left(t \right)}$ .
    
    Then let $du = - \sin{\left(t \right)} dt$ 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:
    
    $- \frac{\cos^{3}{\left(t \right)}}{3}$
  So, the result is: $- \frac{2 \cos^{3}{\left(t \right)}}{3}$
Method #2
1. Rewrite the integrand:
  
  $\sin{\left(2 t \right)} \cos{\left(t \right)} = 2 \sin{\left(t \right)} \cos^{2}{\left(t \right)}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 2 \sin{\left(t \right)} \cos^{2}{\left(t \right)}\, dt = 2 \int \sin{\left(t \right)} \cos^{2}{\left(t \right)}\, dt$
  1. Let $u = \cos{\left(t \right)}$ .
    
    Then let $du = - \sin{\left(t \right)} dt$ 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:
    
    $- \frac{\cos^{3}{\left(t \right)}}{3}$
  So, the result is: $- \frac{2 \cos^{3}{\left(t \right)}}{3}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                              3   
 |                          2*cos (t)
 | sin(2*t)*cos(t) dt = C - ---------
 |                              3    
/

$$\int \sin{\left(2 t \right)} \cos{\left(t \right)}\, dt = C - \frac{2 \cos^{3}{\left(t \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

4/3

$$\frac{4}{3}$$

=

4/3

$$\frac{4}{3}$$

4/3

Numerical answer [src]

1.33333333333333

1.33333333333333

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Integral of sin2t*cost dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2