Integral of sen4x dx

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  1            
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 |  sin(4*x) dx
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0

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

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

Detail solution

Let $u = 4 x$ .

Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :

$\int \frac{\sin{\left(u \right)}}{4}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $- \frac{\cos{\left(u \right)}}{4}$
Now substitute $u$ back in:

$- \frac{\cos{\left(4 x \right)}}{4}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                          
 |                   cos(4*x)
 | sin(4*x) dx = C - --------
 |                      4    
/

\int \sin{\left(4 x \right)}\, dx = C - \frac{\cos{\left(4 x \right)}}{4}

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   cos(4)
- - ------
4     4

\frac{1}{4} - \frac{\cos{\left(4 \right)}}{4}

1   cos(4)
- - ------
4     4

\frac{1}{4} - \frac{\cos{\left(4 \right)}}{4}

1/4 - cos(4)/4

Numerical answer [src]

0.413410905215903

0.413410905215903

The graph

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