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Integral of d{x}:
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Integral of -x^3
Identical expressions
cos3x(sinx)^ two
co sinus of e of 3x( sinus of x) squared
co sinus of e of 3x( sinus of x) to the power of two
cos3x(sinx)2
cos3xsinx2
cos3x(sinx)²
cos3x(sinx) to the power of 2
cos3xsinx^2
cos3x(sinx)^2dx

Solving integrals/ cos3x(sinx)^2

Integral of cos3x(sinx)^2 dx

The solution

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

$$\int\limits_{1}^{3} \sin^{2}{\left(x \right)} \cos{\left(3 x \right)}\, dx$$

Integral(cos(3*x)*sin(x)^2, (x, 1, 3))

Detail solution

Rewrite the integrand:

$\sin^{2}{\left(x \right)} \cos{\left(3 x \right)} = 4 \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)} - 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 4 \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}\, dx = 4 \int \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}\, dx$
  1. Rewrite the integrand:
    
    $\sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
  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 $du$ :
      
      $\int \left(- u^{4} + u^{2}\right)\, du$
      
      Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{4}\right)\, du = - \int u^{4}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
      
      So, the result is: $- \frac{u^{5}}{5}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      The result is: $- \frac{u^{5}}{5} + \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $- \frac{\sin^{5}{\left(x \right)}}{5} + \frac{\sin^{3}{\left(x \right)}}{3}$
    Method #2
    1. Rewrite the integrand:
      
      $\left(1 - \sin^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)} = - \sin^{4}{\left(x \right)} \cos{\left(x \right)} + \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$
      
      Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{4}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{5}{\left(x \right)}}{5}$
      
      So, the result is: $- \frac{\sin^{5}{\left(x \right)}}{5}$
      
      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:
      
      $\frac{\sin^{3}{\left(x \right)}}{3}$
      
      The result is: $- \frac{\sin^{5}{\left(x \right)}}{5} + \frac{\sin^{3}{\left(x \right)}}{3}$
    Method #3
    1. Rewrite the integrand:
      
      $\left(1 - \sin^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)} = - \sin^{4}{\left(x \right)} \cos{\left(x \right)} + \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$
      
      Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{4}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{4}\, du = \frac{u^{5}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{5}{\left(x \right)}}{5}$
      
      So, the result is: $- \frac{\sin^{5}{\left(x \right)}}{5}$
      
      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:
      
      $\frac{\sin^{3}{\left(x \right)}}{3}$
      
      The result is: $- \frac{\sin^{5}{\left(x \right)}}{5} + \frac{\sin^{3}{\left(x \right)}}{3}$
  So, the result is: $- \frac{4 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 3 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int u^{2}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{3}{\left(x \right)}}{3}$
  So, the result is: $- \sin^{3}{\left(x \right)}$
The result is: $- \frac{4 \sin^{5}{\left(x \right)}}{5} + \frac{\sin^{3}{\left(x \right)}}{3}$
Now simplify:

$\frac{\left(5 - 12 \sin^{2}{\left(x \right)}\right) \sin^{3}{\left(x \right)}}{15}$
Add the constant of integration:

$\frac{\left(5 - 12 \sin^{2}{\left(x \right)}\right) \sin^{3}{\left(x \right)}}{15}+ \mathrm{constant}$

The answer is:

$\frac{\left(5 - 12 \sin^{2}{\left(x \right)}\right) \sin^{3}{\left(x \right)}}{15}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                             
 |                                5         3   
 |             2             4*sin (x)   sin (x)
 | cos(3*x)*sin (x) dx = C - --------- + -------
 |                               5          3   
/

$$-{{3\,\sin ^5x-5\,\sin ^3x}\over{15}}-{{3\,\sin ^5x}\over{5}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       2                  2                  2                  2                                                            
  7*sin (1)*sin(3)   2*cos (3)*sin(9)   2*cos (1)*sin(3)   7*sin (3)*sin(9)   2*cos(1)*cos(3)*sin(1)   2*cos(3)*cos(9)*sin(3)
- ---------------- - ---------------- + ---------------- + ---------------- - ---------------------- + ----------------------
         15                 15                 15                 15                    5                        5

$${{12\,\sin ^51-5\,\sin ^31}\over{15}}-{{12\,\sin ^53-5\,\sin ^33 }\over{15}}$$

       2                  2                  2                  2                                                            
  7*sin (1)*sin(3)   2*cos (3)*sin(9)   2*cos (1)*sin(3)   7*sin (3)*sin(9)   2*cos(1)*cos(3)*sin(1)   2*cos(3)*cos(9)*sin(3)
- ---------------- - ---------------- + ---------------- + ---------------- - ---------------------- + ----------------------
         15                 15                 15                 15                    5                        5

$$- \frac{2 \sin{\left(9 \right)} \cos^{2}{\left(3 \right)}}{15} - \frac{7 \sin^{2}{\left(1 \right)} \sin{\left(3 \right)}}{15} + \frac{7 \sin^{2}{\left(3 \right)} \sin{\left(9 \right)}}{15} + \frac{2 \sin{\left(3 \right)} \cos^{2}{\left(1 \right)}}{15} + \frac{2 \sin{\left(3 \right)} \cos{\left(3 \right)} \cos{\left(9 \right)}}{5} - \frac{2 \sin{\left(1 \right)} \cos{\left(1 \right)} \cos{\left(3 \right)}}{5}$$

Simplify

Numerical answer [src]

0.139793551309643

0.139793551309643

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of cos3x(sinx)^2 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3