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Solving integrals/ (cos3x-2x^3)

Integral of (cos3x-2x^3) dx

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The solution

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 |  \cos(3*x) - 2*x / dx
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∫01(−2x3+cos⁡(3x)) dx\int\limits_{0}^{1} \left(- 2 x^{3} + \cos{\left(3 x \right)}\right)\, dx 
Integral(cos(3*x) - 2*x^3, (x, 0, 1))
Detail solution

  1. Integrate term-by-term:

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

      ∫(−2x3) dx=−2∫x3 dx\int \left(- 2 x^{3}\right)\, dx = - 2 \int x^{3}\, dx

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫x3 dx=x44\int x^{3}\, dx = \frac{x^{4}}{4}

      So, the result is: −x42- \frac{x^{4}}{2}

    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}

    The result is: −x42+sin⁡(3x)3- \frac{x^{4}}{2} + \frac{\sin{\left(3 x \right)}}{3}

  2. Add the constant of integration:

    −x42+sin⁡(3x)3+constant- \frac{x^{4}}{2} + \frac{\sin{\left(3 x \right)}}{3}+ \mathrm{constant}

The answer is:

−x42+sin⁡(3x)3+constant- \frac{x^{4}}{2} + \frac{\sin{\left(3 x \right)}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                        
 |                             4           
 | /              3\          x    sin(3*x)
 | \cos(3*x) - 2*x / dx = C - -- + --------
 |                            2       3    
/
∫(−2x3+cos⁡(3x)) dx=C−x42+sin⁡(3x)3\int \left(- 2 x^{3} + \cos{\left(3 x \right)}\right)\, dx = C - \frac{x^{4}}{2} + \frac{\sin{\left(3 x \right)}}{3}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.905-5
Plot the graph f(x)
The answer [src] 
  1   sin(3)
- - + ------
  2     3
−12+sin⁡(3)3- \frac{1}{2} + \frac{\sin{\left(3 \right)}}{3} 
=
= 
  1   sin(3)
- - + ------
  2     3
−12+sin⁡(3)3- \frac{1}{2} + \frac{\sin{\left(3 \right)}}{3} 
-1/2 + sin(3)/3
Numerical answer [src] 
-0.452959997313378
-0.452959997313378

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

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