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Solving integrals/ sqrt2-2sin(x)

Integral of sqrt2-2sin(x) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
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∫0π3(−2sin⁡(x)+2) dx\int\limits_{0}^{\frac{\pi}{3}} \left(- 2 \sin{\left(x \right)} + \sqrt{2}\right)\, dx 
Integral(sqrt(2) - 2*sin(x), (x, 0, pi/3))
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:

      ∫(−2sin⁡(x)) dx=−2∫sin⁡(x) dx\int \left(- 2 \sin{\left(x \right)}\right)\, dx = - 2 \int \sin{\left(x \right)}\, dx

      1. The integral of sine is negative cosine:

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

      So, the result is: 2cos⁡(x)2 \cos{\left(x \right)}

    1. The integral of a constant is the constant times the variable of integration:

      ∫2 dx=2x\int \sqrt{2}\, dx = \sqrt{2} x

    The result is: 2x+2cos⁡(x)\sqrt{2} x + 2 \cos{\left(x \right)}

  2. Add the constant of integration:

    2x+2cos⁡(x)+constant\sqrt{2} x + 2 \cos{\left(x \right)}+ \mathrm{constant}

The answer is:

2x+2cos⁡(x)+constant\sqrt{2} x + 2 \cos{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
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 | \\/ 2  - 2*sin(x)/ dx = C + 2*cos(x) + x*\/ 2 
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∫(−2sin⁡(x)+2) dx=C+2x+2cos⁡(x)\int \left(- 2 \sin{\left(x \right)} + \sqrt{2}\right)\, dx = C + \sqrt{2} x + 2 \cos{\left(x \right)}
Simplify
The graph
0.000.100.200.300.400.500.600.700.800.901.005-5
Plot the graph f(x)
The answer [src] 
          ___
     pi*\/ 2 
-1 + --------
        3
−1+2π3-1 + \frac{\sqrt{2} \pi}{3} 
=
= 
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     pi*\/ 2 
-1 + --------
        3
−1+2π3-1 + \frac{\sqrt{2} \pi}{3} 
-1 + pi*sqrt(2)/3
Numerical answer [src] 
0.480960979386122
0.480960979386122

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

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