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Solving integrals/ cos(3x+5)

Integral of cos(3x+5) dx

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∫00cos⁡(3x+5) dx\int\limits_{0}^{0} \cos{\left(3 x + 5 \right)}\, dx 
Integral(cos(3*x + 5), (x, 0, 0))
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}
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    Use the examples entering the upper and lower limits of integration.

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