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cos^2(5x)
Solving integrals/ cos^2(5x)

Integral of cos^2(5x) dx

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The solution

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∫01cos⁡2(5x) dx\int\limits_{0}^{1} \cos^{2}{\left(5 x \right)}\, dx 
Integral(cos(5*x)^2, (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    cos⁡2(5x)=cos⁡(10x)2+12\cos^{2}{\left(5 x \right)} = \frac{\cos{\left(10 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⁡(10x)2 dx=∫cos⁡(10x) dx2\int \frac{\cos{\left(10 x \right)}}{2}\, dx = \frac{\int \cos{\left(10 x \right)}\, dx}{2}

      1. Let u=10xu = 10 x.

        Then let du=10dxdu = 10 dx and substitute du10\frac{du}{10}:

        ∫cos⁡(u)100 du\int \frac{\cos{\left(u \right)}}{100}\, du

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

          ∫cos⁡(u)10 du=∫cos⁡(u) du10\int \frac{\cos{\left(u \right)}}{10}\, du = \frac{\int \cos{\left(u \right)}\, du}{10}

          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)10\frac{\sin{\left(u \right)}}{10}

        Now substitute uu back in:

        sin⁡(10x)10\frac{\sin{\left(10 x \right)}}{10}

      So, the result is: sin⁡(10x)20\frac{\sin{\left(10 x \right)}}{20}

    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⁡(10x)20\frac{x}{2} + \frac{\sin{\left(10 x \right)}}{20}

  3. Add the constant of integration:

    x2+sin⁡(10x)20+constant\frac{x}{2} + \frac{\sin{\left(10 x \right)}}{20}+ \mathrm{constant}

The answer is:

x2+sin⁡(10x)20+constant\frac{x}{2} + \frac{\sin{\left(10 x \right)}}{20}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                
 |                                 
 |    2               x   sin(10*x)
 | cos (5*x) dx = C + - + ---------
 |                    2       20   
/
sin⁡(10 x)2+5 x10{{{{\sin \left(10\,x\right)}\over{2}}+5\,x}\over{10}}
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(5)*sin(5)
- + -------------
2         10
sin⁡10+1020{{\sin 10+10}\over{20}} 
=
= 
1   cos(5)*sin(5)
- + -------------
2         10
sin⁡(5)cos⁡(5)10+12\frac{\sin{\left(5 \right)} \cos{\left(5 \right)}}{10} + \frac{1}{2}
Simplify
Numerical answer [src] 
0.472798944455532
0.472798944455532
The graph
Integral of cos^2(5x) dx

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

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