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Solving integrals/ cos(x)+sin(3x)

Integral of cos(x)+sin(3x) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
 pi                       
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 |  (cos(x) + sin(3*x)) dx
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∫0π(sin⁡(3x)+cos⁡(x)) dx\int\limits_{0}^{\pi} \left(\sin{\left(3 x \right)} + \cos{\left(x \right)}\right)\, dx 
Integral(cos(x) + sin(3*x), (x, 0, pi))
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}:

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

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

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

        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)3- \frac{\cos{\left(u \right)}}{3}

      Now substitute uu back in:

      −cos⁡(3x)3- \frac{\cos{\left(3 x \right)}}{3}

    1. The integral of cosine is sine:

      ∫cos⁡(x) dx=sin⁡(x)\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}

    The result is: sin⁡(x)−cos⁡(3x)3\sin{\left(x \right)} - \frac{\cos{\left(3 x \right)}}{3}

  2. Add the constant of integration:

    sin⁡(x)−cos⁡(3x)3+constant\sin{\left(x \right)} - \frac{\cos{\left(3 x \right)}}{3}+ \mathrm{constant}

The answer is:

sin⁡(x)−cos⁡(3x)3+constant\sin{\left(x \right)} - \frac{\cos{\left(3 x \right)}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                              
 |                              cos(3*x)         
 | (cos(x) + sin(3*x)) dx = C - -------- + sin(x)
 |                                 3             
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∫(sin⁡(3x)+cos⁡(x)) dx=C+sin⁡(x)−cos⁡(3x)3\int \left(\sin{\left(3 x \right)} + \cos{\left(x \right)}\right)\, dx = C + \sin{\left(x \right)} - \frac{\cos{\left(3 x \right)}}{3}
Simplify
The graph
0.000.250.500.751.001.251.501.752.002.252.502.753.005-5
Plot the graph f(x)
The answer [src] 
2/3
23\frac{2}{3} 
=
= 
2/3
23\frac{2}{3} 
2/3
Numerical answer [src] 
0.666666666666667
0.666666666666667

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

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