Integral of sin(5*x) dx

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

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

Detail solution

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:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                          
 |                   cos(5*x)
 | sin(5*x) dx = C - --------
 |                      5    
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   cos(5)
- - ------
5     5

$${{1}\over{5}}-{{\cos 5}\over{5}}$$

1   cos(5)
- - ------
5     5

$$\frac{1}{5} - \frac{\cos{\left(5 \right)}}{5}$$

Упростить

Numerical answer [src]

0.143267562907355

0.143267562907355

The graph

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