Integral of 4sin5x dx

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 |  4*sin(5*x) dx
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$$\int\limits_{0}^{1} 4 \sin{\left(5 x \right)}\, dx$$

Integral(4*sin(5*x), (x, 0, 1))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int 4 \sin{\left(5 x \right)}\, dx = 4 \int \sin{\left(5 x \right)}\, dx$
1. Let $u = 5 x$ .
  
  Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
  
  $\int \frac{\sin{\left(u \right)}}{25}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sin{\left(u \right)}}{5}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{5}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(5 x \right)}}{5}$
So, the result is: $- \frac{4 \cos{\left(5 x \right)}}{5}$
Add the constant of integration:

$- \frac{4 \cos{\left(5 x \right)}}{5}+ \mathrm{constant}$

The answer is:

$- \frac{4 \cos{\left(5 x \right)}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                              
 |                     4*cos(5*x)
 | 4*sin(5*x) dx = C - ----------
 |                         5     
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$$-{{4\,\cos \left(5\,x\right)}\over{5}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

4   4*cos(5)
- - --------
5      5

$$4\,\left({{1}\over{5}}-{{\cos 5}\over{5}}\right)$$

4   4*cos(5)
- - --------
5      5

$$- \frac{4 \cos{\left(5 \right)}}{5} + \frac{4}{5}$$

Simplify

Numerical answer [src]

0.573070251629419

0.573070251629419

The graph

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