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cos(3x+5)dx
Solving integrals/ cos(3x+5)dx

Integral of cos(3x+5)dx dx

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The solution

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∫01cos⁡(3x+5) dx\int\limits_{0}^{1} \cos{\left(3 x + 5 \right)}\, dx 
Integral(cos(3*x + 5), (x, 0, 1))
Detail solution

  1. Let u=3x+5u = 3 x + 5.

    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+5)3\frac{\sin{\left(3 x + 5 \right)}}{3}

  2. Now simplify:

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                  
 |                       sin(3*x + 5)
 | cos(3*x + 5) dx = C + ------------
 |                            3      
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∫cos⁡(3x+5) dx=C+sin⁡(3x+5)3\int \cos{\left(3 x + 5 \right)}\, dx = C + \frac{\sin{\left(3 x + 5 \right)}}{3}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902-1
Plot the graph f(x)
The answer [src] 
  sin(5)   sin(8)
- ------ + ------
    3        3
−sin⁡(5)3+sin⁡(8)3- \frac{\sin{\left(5 \right)}}{3} + \frac{\sin{\left(8 \right)}}{3} 
=
= 
  sin(5)   sin(8)
- ------ + ------
    3        3
−sin⁡(5)3+sin⁡(8)3- \frac{\sin{\left(5 \right)}}{3} + \frac{\sin{\left(8 \right)}}{3} 
-sin(5)/3 + sin(8)/3
Numerical answer [src] 
0.649427507095507
0.649427507095507
The graph
Integral of cos(3x+5)dx dx

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

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