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cos²3x
Solving integrals/ cos²3x

Integral of cos²3x dx

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The solution

You have entered [src] 
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∫01cos⁡2(3x) dx\int\limits_{0}^{1} \cos^{2}{\left(3 x \right)}\, dx 
Integral(cos(3*x)^2, (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    cos⁡2(3x)=cos⁡(6x)2+12\cos^{2}{\left(3 x \right)} = \frac{\cos{\left(6 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⁡(6x)2 dx=∫cos⁡(6x) dx2\int \frac{\cos{\left(6 x \right)}}{2}\, dx = \frac{\int \cos{\left(6 x \right)}\, dx}{2}

      1. Let u=6xu = 6 x.

        Then let du=6dxdu = 6 dx and substitute du6\frac{du}{6}:

        ∫cos⁡(u)6 du\int \frac{\cos{\left(u \right)}}{6}\, du

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

          ∫cos⁡(u) du=∫cos⁡(u) du6\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{6}

          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)6\frac{\sin{\left(u \right)}}{6}

        Now substitute uu back in:

        sin⁡(6x)6\frac{\sin{\left(6 x \right)}}{6}

      So, the result is: sin⁡(6x)12\frac{\sin{\left(6 x \right)}}{12}

    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⁡(6x)12\frac{x}{2} + \frac{\sin{\left(6 x \right)}}{12}

  3. Add the constant of integration:

    x2+sin⁡(6x)12+constant\frac{x}{2} + \frac{\sin{\left(6 x \right)}}{12}+ \mathrm{constant}

The answer is:

x2+sin⁡(6x)12+constant\frac{x}{2} + \frac{\sin{\left(6 x \right)}}{12}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                               
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 |    2               x   sin(6*x)
 | cos (3*x) dx = C + - + --------
 |                    2      12   
/
∫cos⁡2(3x) dx=C+x2+sin⁡(6x)12\int \cos^{2}{\left(3 x \right)}\, dx = C + \frac{x}{2} + \frac{\sin{\left(6 x \right)}}{12}
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(3)*sin(3)
- + -------------
2         6
sin⁡(3)cos⁡(3)6+12\frac{\sin{\left(3 \right)} \cos{\left(3 \right)}}{6} + \frac{1}{2} 
=
= 
1   cos(3)*sin(3)
- + -------------
2         6
sin⁡(3)cos⁡(3)6+12\frac{\sin{\left(3 \right)} \cos{\left(3 \right)}}{6} + \frac{1}{2} 
1/2 + cos(3)*sin(3)/6
Numerical answer [src] 
0.476715375150089
0.476715375150089
The graph
Integral of cos²3x dx

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

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