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Integral of d{x}:
Integral of 4x³
Integral of sec^2xtanx
Integral of chx
Integral of sin^3(5x)×cos^4(5x)
Identical expressions
sin^ three (5x)×cos^ four (5x)
sinus of cubed (5x)× co sinus of e of to the power of 4(5x)
sinus of to the power of three (5x)× co sinus of e of to the power of four (5x)
sin3(5x)×cos4(5x)
sin35x×cos45x
sin³(5x)×cos⁴(5x)
sin to the power of 3(5x)×cos to the power of 4(5x)
sin^35x×cos^45x
sin^3(5x)×cos^4(5x)dx

Solving integrals/ sin^3(5x)×cos^4(5x)

Integral of sin^3(5x)×cos^4(5x) dx

The solution

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  0                       
  /                       
 |                        
 |     3         4        
 |  sin (5*x)*cos (5*x) dx
 |                        
/                         
0

$$\int\limits_{0}^{0} \sin^{3}{\left(5 x \right)} \cos^{4}{\left(5 x \right)}\, dx$$

Integral(sin(5*x)^3*cos(5*x)^4, (x, 0, 0))

Detail solution

Rewrite the integrand:

$\sin^{3}{\left(5 x \right)} \cos^{4}{\left(5 x \right)} = \left(1 - \cos^{2}{\left(5 x \right)}\right) \sin{\left(5 x \right)} \cos^{4}{\left(5 x \right)}$
There are multiple ways to do this integral.
Method #1
1. Let $u = \cos{\left(5 x \right)}$ .
  
  Then let $du = - 5 \sin{\left(5 x \right)} dx$ and substitute $du$ :
  
  $\int \left(\frac{u^{6}}{5} - \frac{u^{4}}{5}\right)\, du$
  1. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{u^{6}}{5}\, du = \frac{\int u^{6}\, du}{5}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
      
      So, the result is: $\frac{u^{7}}{35}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{u^{4}}{5}\right)\, du = - \frac{\int u^{4}\, du}{5}$
      
      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: $- \frac{u^{5}}{25}$
    The result is: $\frac{u^{7}}{35} - \frac{u^{5}}{25}$
  Now substitute $u$ back in:
  
  $\frac{\cos^{7}{\left(5 x \right)}}{35} - \frac{\cos^{5}{\left(5 x \right)}}{25}$
Method #2
1. Rewrite the integrand:
  
  $\left(1 - \cos^{2}{\left(5 x \right)}\right) \sin{\left(5 x \right)} \cos^{4}{\left(5 x \right)} = - \sin{\left(5 x \right)} \cos^{6}{\left(5 x \right)} + \sin{\left(5 x \right)} \cos^{4}{\left(5 x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin{\left(5 x \right)} \cos^{6}{\left(5 x \right)}\right)\, dx = - \int \sin{\left(5 x \right)} \cos^{6}{\left(5 x \right)}\, dx$
    1. Let $u = \cos{\left(5 x \right)}$ .
      
      Then let $du = - 5 \sin{\left(5 x \right)} dx$ and substitute $- \frac{du}{5}$ :
      
      $\int \frac{u^{6}}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{u^{6}}{5}\right)\, du = - \frac{\int u^{6}\, du}{5}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
      
      So, the result is: $- \frac{u^{7}}{35}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{7}{\left(5 x \right)}}{35}$
    So, the result is: $\frac{\cos^{7}{\left(5 x \right)}}{35}$
  1. Let $u = \cos{\left(5 x \right)}$ .
    
    Then let $du = - 5 \sin{\left(5 x \right)} dx$ and substitute $- \frac{du}{5}$ :
    
    $\int \frac{u^{4}}{25}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{u^{4}}{5}\right)\, du = - \frac{\int u^{4}\, du}{5}$
      
      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: $- \frac{u^{5}}{25}$
    Now substitute $u$ back in:
    
    $- \frac{\cos^{5}{\left(5 x \right)}}{25}$
  The result is: $\frac{\cos^{7}{\left(5 x \right)}}{35} - \frac{\cos^{5}{\left(5 x \right)}}{25}$
Method #3
1. Rewrite the integrand:
  
  $\left(1 - \cos^{2}{\left(5 x \right)}\right) \sin{\left(5 x \right)} \cos^{4}{\left(5 x \right)} = - \sin{\left(5 x \right)} \cos^{6}{\left(5 x \right)} + \sin{\left(5 x \right)} \cos^{4}{\left(5 x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin{\left(5 x \right)} \cos^{6}{\left(5 x \right)}\right)\, dx = - \int \sin{\left(5 x \right)} \cos^{6}{\left(5 x \right)}\, dx$
    1. Let $u = \cos{\left(5 x \right)}$ .
      
      Then let $du = - 5 \sin{\left(5 x \right)} dx$ and substitute $- \frac{du}{5}$ :
      
      $\int \frac{u^{6}}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{u^{6}}{5}\right)\, du = - \frac{\int u^{6}\, du}{5}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
      
      So, the result is: $- \frac{u^{7}}{35}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{7}{\left(5 x \right)}}{35}$
    So, the result is: $\frac{\cos^{7}{\left(5 x \right)}}{35}$
  1. Let $u = \cos{\left(5 x \right)}$ .
    
    Then let $du = - 5 \sin{\left(5 x \right)} dx$ and substitute $- \frac{du}{5}$ :
    
    $\int \frac{u^{4}}{25}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{u^{4}}{5}\right)\, du = - \frac{\int u^{4}\, du}{5}$
      
      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: $- \frac{u^{5}}{25}$
    Now substitute $u$ back in:
    
    $- \frac{\cos^{5}{\left(5 x \right)}}{25}$
  The result is: $\frac{\cos^{7}{\left(5 x \right)}}{35} - \frac{\cos^{5}{\left(5 x \right)}}{25}$
Now simplify:

$\frac{\left(5 \cos{\left(10 x \right)} - 9\right) \cos^{5}{\left(5 x \right)}}{350}$
Add the constant of integration:

$\frac{\left(5 \cos{\left(10 x \right)} - 9\right) \cos^{5}{\left(5 x \right)}}{350}+ \mathrm{constant}$

The answer is:

$\frac{\left(5 \cos{\left(10 x \right)} - 9\right) \cos^{5}{\left(5 x \right)}}{350}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                  
 |                                 5           7     
 |    3         4               cos (5*x)   cos (5*x)
 | sin (5*x)*cos (5*x) dx = C - --------- + ---------
 |                                  25          35   
/

$${{5\,\cos ^7\left(5\,x\right)-7\,\cos ^5\left(5\,x\right)}\over{175 }}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Simplify

Numerical answer [src]

0.0

0.0

The graph

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

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Identical expressions

Integral of sin^3(5x)×cos^4(5x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3