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(sin4x+1)dx
Solving integrals/ (sin4x+1)dx

Integral of (sin4x+1)dx dx

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The solution

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 |  (sin(4*x) + 1) dx
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∫01(sin⁡(4x)+1) dx\int\limits_{0}^{1} \left(\sin{\left(4 x \right)} + 1\right)\, dx 
Integral(sin(4*x) + 1, (x, 0, 1))
Detail solution

  1. Integrate term-by-term:

    1. Let u=4xu = 4 x.

      Then let du=4dxdu = 4 dx and substitute du4\frac{du}{4}:

      ∫sin⁡(u)4 du\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:

        ∫sin⁡(u) du=∫sin⁡(u) du4\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}

        1. The integral of sine is negative cosine:

          ∫sin⁡(u) du=−cos⁡(u)\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}

        So, the result is: −cos⁡(u)4- \frac{\cos{\left(u \right)}}{4}

      Now substitute uu back in:

      −cos⁡(4x)4- \frac{\cos{\left(4 x \right)}}{4}

    1. The integral of a constant is the constant times the variable of integration:

      ∫1 dx=x\int 1\, dx = x

    The result is: x−cos⁡(4x)4x - \frac{\cos{\left(4 x \right)}}{4}

  2. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                    
 |                             cos(4*x)
 | (sin(4*x) + 1) dx = C + x - --------
 |                                4    
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∫(sin⁡(4x)+1) dx=C+x−cos⁡(4x)4\int \left(\sin{\left(4 x \right)} + 1\right)\, dx = C + x - \frac{\cos{\left(4 x \right)}}{4}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902.5-2.5
Plot the graph f(x)
The answer [src] 
5   cos(4)
- - ------
4     4
54−cos⁡(4)4\frac{5}{4} - \frac{\cos{\left(4 \right)}}{4} 
=
= 
5   cos(4)
- - ------
4     4
54−cos⁡(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
Integral of (sin4x+1)dx dx

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

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