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1/1-(cos(x))^2
Solving integrals/ 1/1-(cos(x))^2

Integral of 1/1-(cos(x))^2 dx

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∫π4π2(1−cos⁡2(x)) dx\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{2}} \left(1 - \cos^{2}{\left(x \right)}\right)\, dx 
Integral(1 - cos(x)^2, (x, pi/4, pi/2))
Detail solution

  1. Integrate term-by-term:

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

      ∫1 dx=x\int 1\, dx = x

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

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

      1. Rewrite the integrand:

        cos⁡2(x)=cos⁡(2x)2+12\cos^{2}{\left(x \right)} = \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}

      2. Integrate term-by-term:

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

          ∫cos⁡(2x)2 dx=∫cos⁡(2x) dx2\int \frac{\cos{\left(2 x \right)}}{2}\, dx = \frac{\int \cos{\left(2 x \right)}\, dx}{2}

          1. Let u=2xu = 2 x.

            Then let du=2dxdu = 2 dx and substitute du2\frac{du}{2}:

            ∫cos⁡(u)4 du\int \frac{\cos{\left(u \right)}}{4}\, du

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

              ∫cos⁡(u)2 du=∫cos⁡(u) du2\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}

              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)2\frac{\sin{\left(u \right)}}{2}

            Now substitute uu back in:

            sin⁡(2x)2\frac{\sin{\left(2 x \right)}}{2}

          So, the result is: sin⁡(2x)4\frac{\sin{\left(2 x \right)}}{4}

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

          ∫12 dx=x2\int \frac{1}{2}\, dx = \frac{x}{2}

        The result is: x2+sin⁡(2x)4\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}

      So, the result is: −x2−sin⁡(2x)4- \frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}

    The result is: x2−sin⁡(2x)4\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}

  2. Add the constant of integration:

    x2−sin⁡(2x)4+constant\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}

The answer is:

x2−sin⁡(2x)4+constant\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                   
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 | /       2   \          x   sin(2*x)
 | \1 - cos (x)/ dx = C + - - --------
 |                        2      4    
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∫(1−cos⁡2(x)) dx=C+x2−sin⁡(2x)4\int \left(1 - \cos^{2}{\left(x \right)}\right)\, dx = C + \frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}
Simplify
The graph
0.800.850.900.951.001.051.101.151.201.251.301.351.401.451.501.550.02.0
Plot the graph f(x)
The answer [src] 
1   pi
- + --
4   8
14+π8\frac{1}{4} + \frac{\pi}{8} 
=
= 
1   pi
- + --
4   8
14+π8\frac{1}{4} + \frac{\pi}{8}
Упростить
Numerical answer [src] 
0.642699081698724
0.642699081698724
The graph
Integral of 1/1-(cos(x))^2 dx

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

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