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Solving integrals/ sqrt(3)*cos(x)-sin(x)

Integral of sqrt(3)*cos(x)-sin(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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∫0π2(−sin⁡(x)+3cos⁡(x)) dx\int\limits_{0}^{\frac{\pi}{2}} \left(- \sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)}\right)\, dx 
Integral(sqrt(3)*cos(x) - sin(x), (x, 0, pi/2))
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:

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

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

      1. The integral of cosine is sine:

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

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

    The result is: 3sin⁡(x)+cos⁡(x)\sqrt{3} \sin{\left(x \right)} + \cos{\left(x \right)}

  2. Now simplify:

    2sin⁡(x+π6)2 \sin{\left(x + \frac{\pi}{6} \right)}

  3. Add the constant of integration:

    2sin⁡(x+π6)+constant2 \sin{\left(x + \frac{\pi}{6} \right)}+ \mathrm{constant}

The answer is:

2sin⁡(x+π6)+constant2 \sin{\left(x + \frac{\pi}{6} \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
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 | \\/ 3 *cos(x) - sin(x)/ dx = C + \/ 3 *sin(x) + cos(x)
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∫(−sin⁡(x)+3cos⁡(x)) dx=C+3sin⁡(x)+cos⁡(x)\int \left(- \sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)}\right)\, dx = C + \sqrt{3} \sin{\left(x \right)} + \cos{\left(x \right)}
Simplify
The graph
0.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.55-5
Plot the graph f(x)
The answer [src] 
       ___
-1 + \/ 3
−1+3-1 + \sqrt{3} 
=
= 
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-1 + \/ 3
−1+3-1 + \sqrt{3} 
-1 + sqrt(3)
Numerical answer [src] 
0.732050807568877
0.732050807568877

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

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