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Solving integrals/ 3cosx/3-x

Integral of 3cosx/3-x dx

Limits of integration:

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Piecewise:

The solution

You have entered [src] 
  1                  
  /                  
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 |  /3*cos(x)    \   
 |  |-------- - x| dx
 |  \   3        /   
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0
∫01(−x+3cos⁡(x)3) dx\int\limits_{0}^{1} \left(- x + \frac{3 \cos{\left(x \right)}}{3}\right)\, dx 
Integral((3*cos(x))/3 - x, (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:

      ∫(−x) dx=−∫x dx\int \left(- x\right)\, dx = - \int x\, dx

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

        ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

      So, the result is: −x22- \frac{x^{2}}{2}

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

      ∫3cos⁡(x)3 dx=∫3cos⁡(x) dx3\int \frac{3 \cos{\left(x \right)}}{3}\, dx = \frac{\int 3 \cos{\left(x \right)}\, dx}{3}

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

        ∫3cos⁡(x) dx=3∫cos⁡(x) dx\int 3 \cos{\left(x \right)}\, dx = 3 \int \cos{\left(x \right)}\, dx

        1. Don't know the steps in finding this integral.

          But the integral is

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

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

      So, the result is: sin⁡(x)\sin{\left(x \right)}

    The result is: −x22+sin⁡(x)- \frac{x^{2}}{2} + \sin{\left(x \right)}

  2. Add the constant of integration:

    −x22+sin⁡(x)+constant- \frac{x^{2}}{2} + \sin{\left(x \right)}+ \mathrm{constant}

The answer is:

−x22+sin⁡(x)+constant- \frac{x^{2}}{2} + \sin{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                   
 |                          2         
 | /3*cos(x)    \          x          
 | |-------- - x| dx = C - -- + sin(x)
 | \   3        /          2          
 |                                    
/
∫(−x+3cos⁡(x)3) dx=C−x22+sin⁡(x)\int \left(- x + \frac{3 \cos{\left(x \right)}}{3}\right)\, dx = C - \frac{x^{2}}{2} + \sin{\left(x \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
Plot the graph f(x)
The answer [src] 
-1/2 + sin(1)
−12+sin⁡(1)- \frac{1}{2} + \sin{\left(1 \right)} 
=
= 
-1/2 + sin(1)
−12+sin⁡(1)- \frac{1}{2} + \sin{\left(1 \right)} 
-1/2 + sin(1)
Numerical answer [src] 
0.341470984807897
0.341470984807897

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

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