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Solving integrals/ cos3x+5dx

Integral of cos3x+5dx dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
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 |  (cos(3*x) + 5) dx
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∫01(cos⁡(3x)+5) dx\int\limits_{0}^{1} \left(\cos{\left(3 x \right)} + 5\right)\, dx 
Integral(cos(3*x) + 5, (x, 0, 1))
Detail solution

  1. Integrate term-by-term:

    1. Let u=3xu = 3 x.

      Then let du=3dxdu = 3 dx and substitute du3\frac{du}{3}:

      ∫cos⁡(u)3 du\int \frac{\cos{\left(u \right)}}{3}\, du

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

        ∫cos⁡(u) du=∫cos⁡(u) du3\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{3}

        1. The integral of cosine is sine:

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

        So, the result is: sin⁡(u)3\frac{\sin{\left(u \right)}}{3}

      Now substitute uu back in:

      sin⁡(3x)3\frac{\sin{\left(3 x \right)}}{3}

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

      ∫5 dx=5x\int 5\, dx = 5 x

    The result is: 5x+sin⁡(3x)35 x + \frac{\sin{\left(3 x \right)}}{3}

  2. Add the constant of integration:

    5x+sin⁡(3x)3+constant5 x + \frac{\sin{\left(3 x \right)}}{3}+ \mathrm{constant}

The answer is:

5x+sin⁡(3x)3+constant5 x + \frac{\sin{\left(3 x \right)}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                      
 |                               sin(3*x)
 | (cos(3*x) + 5) dx = C + 5*x + --------
 |                                  3    
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∫(cos⁡(3x)+5) dx=C+5x+sin⁡(3x)3\int \left(\cos{\left(3 x \right)} + 5\right)\, dx = C + 5 x + \frac{\sin{\left(3 x \right)}}{3}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90010
Plot the graph f(x)
The answer [src] 
    sin(3)
5 + ------
      3
sin⁡(3)3+5\frac{\sin{\left(3 \right)}}{3} + 5 
=
= 
    sin(3)
5 + ------
      3
sin⁡(3)3+5\frac{\sin{\left(3 \right)}}{3} + 5 
5 + sin(3)/3
Numerical answer [src] 
5.04704000268662
5.04704000268662

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

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