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cos^(2)7x
Solving integrals/ cos^(2)7x

Integral of cos^(2)7x dx

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The solution

You have entered [src] 
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∫01cos⁡2(7x) dx\int\limits_{0}^{1} \cos^{2}{\left(7 x \right)}\, dx 
Integral(cos(7*x)^2, (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    cos⁡2(7x)=cos⁡(14x)2+12\cos^{2}{\left(7 x \right)} = \frac{\cos{\left(14 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⁡(14x)2 dx=∫cos⁡(14x) dx2\int \frac{\cos{\left(14 x \right)}}{2}\, dx = \frac{\int \cos{\left(14 x \right)}\, dx}{2}

      1. Let u=14xu = 14 x.

        Then let du=14dxdu = 14 dx and substitute du14\frac{du}{14}:

        ∫cos⁡(u)196 du\int \frac{\cos{\left(u \right)}}{196}\, du

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

          ∫cos⁡(u)14 du=∫cos⁡(u) du14\int \frac{\cos{\left(u \right)}}{14}\, du = \frac{\int \cos{\left(u \right)}\, du}{14}

          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)14\frac{\sin{\left(u \right)}}{14}

        Now substitute uu back in:

        sin⁡(14x)14\frac{\sin{\left(14 x \right)}}{14}

      So, the result is: sin⁡(14x)28\frac{\sin{\left(14 x \right)}}{28}

    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⁡(14x)28\frac{x}{2} + \frac{\sin{\left(14 x \right)}}{28}

  3. Add the constant of integration:

    x2+sin⁡(14x)28+constant\frac{x}{2} + \frac{\sin{\left(14 x \right)}}{28}+ \mathrm{constant}

The answer is:

x2+sin⁡(14x)28+constant\frac{x}{2} + \frac{\sin{\left(14 x \right)}}{28}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                
 |                                 
 |    2               x   sin(14*x)
 | cos (7*x) dx = C + - + ---------
 |                    2       28   
/
sin⁡(14 x)2+7 x14{{{{\sin \left(14\,x\right)}\over{2}}+7\,x}\over{14}}
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(7)*sin(7)
- + -------------
2         14
sin⁡14+1428{{\sin 14+14}\over{28}} 
=
= 
1   cos(7)*sin(7)
- + -------------
2         14
sin⁡(7)cos⁡(7)14+12\frac{\sin{\left(7 \right)} \cos{\left(7 \right)}}{14} + \frac{1}{2}
Simplify
Numerical answer [src] 
0.53537883413196
0.53537883413196
The graph
Integral of cos^(2)7x dx

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

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