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cos^2x
Solving integrals/ cos^2x

Integral of cos^2x dx

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The solution

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∫01cos⁡2(x) dx\int\limits_{0}^{1} \cos^{2}{\left(x \right)}\, dx 
Integral(cos(x)^2, (x, 0, 1))
Detail solution

  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)2 du\int \frac{\cos{\left(u \right)}}{2}\, du

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

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

  3. 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)
 | cos (x) dx = C + - + --------
 |                  2      4    
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∫cos⁡2(x) dx=C+x2+sin⁡(2x)4\int \cos^{2}{\left(x \right)}\, dx = C + \frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
Plot the graph f(x)
The answer [src] 
1   cos(1)*sin(1)
- + -------------
2         2
sin⁡(1)cos⁡(1)2+12\frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{1}{2} 
=
= 
1   cos(1)*sin(1)
- + -------------
2         2
sin⁡(1)cos⁡(1)2+12\frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{1}{2} 
1/2 + cos(1)*sin(1)/2
Numerical answer [src] 
0.72732435670642
0.72732435670642
The graph
Integral of cos^2x dx

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

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