Integral of (sin4x+1)dx dx

The solution

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  1                  
  /                  
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 |  (sin(4*x) + 1) dx
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0

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

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

Detail solution

Integrate term-by-term:
1. 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:
  
  $- \frac{\cos{\left(4 x \right)}}{4}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
The result is: $x - \frac{\cos{\left(4 x \right)}}{4}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                    
 |                             cos(4*x)
 | (sin(4*x) + 1) dx = C + x - --------
 |                                4    
/

$$\int \left(\sin{\left(4 x \right)} + 1\right)\, dx = C + x - \frac{\cos{\left(4 x \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

5/4 - cos(4)/4

Numerical answer [src]

1.4134109052159

1.4134109052159

The graph

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

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Identical expressions

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Integral of (sin4x+1)dx dx

Limits of integration:

The graph:

Piecewise:

The solution