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Solving integrals/ (2sinx-cosx)dx

Integral of (2sinx-cosx)dx dx

Limits of integration:

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Piecewise:

The solution

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∫π2π2(2sin⁡(x)−cos⁡(x)) dx\int\limits_{\frac{\pi}{2}}^{\frac{\pi}{2}} \left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right)\, dx 
Integral(2*sin(x) - cos(x), (x, pi/2, 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:

      ∫2sin⁡(x) dx=2∫sin⁡(x) dx\int 2 \sin{\left(x \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 times a function is the constant times the integral of the function:

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

    The result is: −sin⁡(x)−2cos⁡(x)- \sin{\left(x \right)} - 2 \cos{\left(x \right)}

  2. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
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 | (2*sin(x) - cos(x)) dx = C - sin(x) - 2*cos(x)
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∫(2sin⁡(x)−cos⁡(x)) dx=C−sin⁡(x)−2cos⁡(x)\int \left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right)\, dx = C - \sin{\left(x \right)} - 2 \cos{\left(x \right)}
Simplify
The graph
1.57101.57201.57301.57401.57501.57601.57701.57801.57901.58005-5
Plot the graph f(x)
The answer [src] 
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Numerical answer [src] 
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    Use the examples entering the upper and lower limits of integration.

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