Integral of sin3ycosydy dx

The solution

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 |  sin(3*y)*cos(y)*1 dy
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$$\int\limits_{0}^{1} \sin{\left(3 y \right)} \cos{\left(y \right)} 1\, dy$$

Simplify

Detail solution

Rewrite the integrand:

$\sin{\left(3 y \right)} \cos{\left(y \right)} 1 = - 4 \sin^{3}{\left(y \right)} \cos{\left(y \right)} + 3 \sin{\left(y \right)} \cos{\left(y \right)}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 4 \sin^{3}{\left(y \right)} \cos{\left(y \right)}\right)\, dy = - 4 \int \sin^{3}{\left(y \right)} \cos{\left(y \right)}\, dy$
  1. There are multiple ways to do this integral.
    Method #1
    1. Let $u = \sin{\left(y \right)}$ .
      
      Then let $du = \cos{\left(y \right)} dy$ and substitute $du$ :
      
      $\int u^{3}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{3}\, du = \frac{u^{4}}{4}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{4}{\left(y \right)}}{4}$
    Method #2
    1. Rewrite the integrand:
      
      $\sin^{3}{\left(y \right)} \cos{\left(y \right)} = \left(1 - \cos^{2}{\left(y \right)}\right) \sin{\left(y \right)} \cos{\left(y \right)}$
    2. Let $u = - \cos^{2}{\left(y \right)}$ .
      
      Then let $du = 2 \sin{\left(y \right)} \cos{\left(y \right)} dy$ and substitute $du$ :
      
      $\int \left(\frac{u}{2} + \frac{1}{2}\right)\, du$
      
      Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{u}{2}\, du = \frac{\int u\, du}{2}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $\frac{u^{2}}{4}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, du = \frac{u}{2}$
      
      The result is: $\frac{u^{2}}{4} + \frac{u}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\cos^{4}{\left(y \right)}}{4} - \frac{\cos^{2}{\left(y \right)}}{2}$
  So, the result is: $- \sin^{4}{\left(y \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 3 \sin{\left(y \right)} \cos{\left(y \right)}\, dy = 3 \int \sin{\left(y \right)} \cos{\left(y \right)}\, dy$
  1. Let $u = \cos{\left(y \right)}$ .
    
    Then let $du = - \sin{\left(y \right)} dy$ and substitute $- du$ :
    
    $\int u\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u\right)\, du = - \int u\, du$
      1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int u\, du = \frac{u^{2}}{2}$
      So, the result is: $- \frac{u^{2}}{2}$
    Now substitute $u$ back in:
    
    $- \frac{\cos^{2}{\left(y \right)}}{2}$
  So, the result is: $- \frac{3 \cos^{2}{\left(y \right)}}{2}$
The result is: $- \sin^{4}{\left(y \right)} - \frac{3 \cos^{2}{\left(y \right)}}{2}$
Add the constant of integration:

$- \sin^{4}{\left(y \right)} - \frac{3 \cos^{2}{\left(y \right)}}{2}+ \mathrm{constant}$

The answer is:

$- \sin^{4}{\left(y \right)} - \frac{3 \cos^{2}{\left(y \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                          2   
 |                               4      3*cos (y)
 | sin(3*y)*cos(y)*1 dy = C - sin (y) - ---------
 |                                          2    
/

$$-{{\cos \left(4\,y\right)}\over{8}}-{{\cos \left(2\,y\right)}\over{ 4}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

3   3*cos(1)*cos(3)   sin(1)*sin(3)
- - --------------- - -------------
8          8                8

$${{3}\over{8}}-{{\cos 4+2\,\cos 2}\over{8}}$$

3   3*cos(1)*cos(3)   sin(1)*sin(3)
- - --------------- - -------------
8          8                8

$$- \frac{\sin{\left(1 \right)} \sin{\left(3 \right)}}{8} - \frac{3 \cos{\left(1 \right)} \cos{\left(3 \right)}}{8} + \frac{3}{8}$$

Simplify

Numerical answer [src]

0.560742161744737

0.560742161744737

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of sin3ycosydy dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2