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cos^37x
Solving integrals/ cos^37x

Integral of cos^37x dx

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∫01cos⁡3(7x) dx\int\limits_{0}^{1} \cos^{3}{\left(7 x \right)}\, dx 
Integral(cos(7*x)^3, (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    cos⁡3(7x)=(1−sin⁡2(7x))cos⁡(7x)\cos^{3}{\left(7 x \right)} = \left(1 - \sin^{2}{\left(7 x \right)}\right) \cos{\left(7 x \right)}

  2. There are multiple ways to do this integral.

    Method #1

    1. Let u=sin⁡(7x)u = \sin{\left(7 x \right)}.

      Then let du=7cos⁡(7x)dxdu = 7 \cos{\left(7 x \right)} dx and substitute dudu:

      ∫(17−u27) du\int \left(\frac{1}{7} - \frac{u^{2}}{7}\right)\, du

      1. Integrate term-by-term:

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

          ∫17 du=u7\int \frac{1}{7}\, du = \frac{u}{7}

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

          ∫(−u27) du=−∫u2 du7\int \left(- \frac{u^{2}}{7}\right)\, du = - \frac{\int u^{2}\, du}{7}

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

          So, the result is: −u321- \frac{u^{3}}{21}

        The result is: −u321+u7- \frac{u^{3}}{21} + \frac{u}{7}

      Now substitute uu back in:

      −sin⁡3(7x)21+sin⁡(7x)7- \frac{\sin^{3}{\left(7 x \right)}}{21} + \frac{\sin{\left(7 x \right)}}{7}

    Method #2

    1. Rewrite the integrand:

      (1−sin⁡2(7x))cos⁡(7x)=−sin⁡2(7x)cos⁡(7x)+cos⁡(7x)\left(1 - \sin^{2}{\left(7 x \right)}\right) \cos{\left(7 x \right)} = - \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)} + \cos{\left(7 x \right)}

    2. Integrate term-by-term:

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

        ∫(−sin⁡2(7x)cos⁡(7x)) dx=−∫sin⁡2(7x)cos⁡(7x) dx\int \left(- \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)}\right)\, dx = - \int \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)}\, dx

        1. Let u=sin⁡(7x)u = \sin{\left(7 x \right)}.

          Then let du=7cos⁡(7x)dxdu = 7 \cos{\left(7 x \right)} dx and substitute du7\frac{du}{7}:

          ∫u249 du\int \frac{u^{2}}{49}\, du

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

            ∫u27 du=∫u2 du7\int \frac{u^{2}}{7}\, du = \frac{\int u^{2}\, du}{7}

            1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

              ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

            So, the result is: u321\frac{u^{3}}{21}

          Now substitute uu back in:

          sin⁡3(7x)21\frac{\sin^{3}{\left(7 x \right)}}{21}

        So, the result is: −sin⁡3(7x)21- \frac{\sin^{3}{\left(7 x \right)}}{21}

      1. Let u=7xu = 7 x.

        Then let du=7dxdu = 7 dx and substitute du7\frac{du}{7}:

        ∫cos⁡(u)49 du\int \frac{\cos{\left(u \right)}}{49}\, du

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

          ∫cos⁡(u)7 du=∫cos⁡(u) du7\int \frac{\cos{\left(u \right)}}{7}\, du = \frac{\int \cos{\left(u \right)}\, du}{7}

          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)7\frac{\sin{\left(u \right)}}{7}

        Now substitute uu back in:

        sin⁡(7x)7\frac{\sin{\left(7 x \right)}}{7}

      The result is: −sin⁡3(7x)21+sin⁡(7x)7- \frac{\sin^{3}{\left(7 x \right)}}{21} + \frac{\sin{\left(7 x \right)}}{7}

    Method #3

    1. Rewrite the integrand:

      (1−sin⁡2(7x))cos⁡(7x)=−sin⁡2(7x)cos⁡(7x)+cos⁡(7x)\left(1 - \sin^{2}{\left(7 x \right)}\right) \cos{\left(7 x \right)} = - \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)} + \cos{\left(7 x \right)}

    2. Integrate term-by-term:

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

        ∫(−sin⁡2(7x)cos⁡(7x)) dx=−∫sin⁡2(7x)cos⁡(7x) dx\int \left(- \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)}\right)\, dx = - \int \sin^{2}{\left(7 x \right)} \cos{\left(7 x \right)}\, dx

        1. Let u=sin⁡(7x)u = \sin{\left(7 x \right)}.

          Then let du=7cos⁡(7x)dxdu = 7 \cos{\left(7 x \right)} dx and substitute du7\frac{du}{7}:

          ∫u249 du\int \frac{u^{2}}{49}\, du

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

            ∫u27 du=∫u2 du7\int \frac{u^{2}}{7}\, du = \frac{\int u^{2}\, du}{7}

            1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

              ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

            So, the result is: u321\frac{u^{3}}{21}

          Now substitute uu back in:

          sin⁡3(7x)21\frac{\sin^{3}{\left(7 x \right)}}{21}

        So, the result is: −sin⁡3(7x)21- \frac{\sin^{3}{\left(7 x \right)}}{21}

      1. Let u=7xu = 7 x.

        Then let du=7dxdu = 7 dx and substitute du7\frac{du}{7}:

        ∫cos⁡(u)49 du\int \frac{\cos{\left(u \right)}}{49}\, du

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

          ∫cos⁡(u)7 du=∫cos⁡(u) du7\int \frac{\cos{\left(u \right)}}{7}\, du = \frac{\int \cos{\left(u \right)}\, du}{7}

          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)7\frac{\sin{\left(u \right)}}{7}

        Now substitute uu back in:

        sin⁡(7x)7\frac{\sin{\left(7 x \right)}}{7}

      The result is: −sin⁡3(7x)21+sin⁡(7x)7- \frac{\sin^{3}{\left(7 x \right)}}{21} + \frac{\sin{\left(7 x \right)}}{7}

  3. Now simplify:

    3sin⁡(7x)28+sin⁡(21x)84\frac{3 \sin{\left(7 x \right)}}{28} + \frac{\sin{\left(21 x \right)}}{84}

  4. Add the constant of integration:

    3sin⁡(7x)28+sin⁡(21x)84+constant\frac{3 \sin{\left(7 x \right)}}{28} + \frac{\sin{\left(21 x \right)}}{84}+ \mathrm{constant}

The answer is:

3sin⁡(7x)28+sin⁡(21x)84+constant\frac{3 \sin{\left(7 x \right)}}{28} + \frac{\sin{\left(21 x \right)}}{84}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                       
 |                       3                
 |    3               sin (7*x)   sin(7*x)
 | cos (7*x) dx = C - --------- + --------
 |                        21         7    
/
sin⁡(7 x)−sin⁡3(7 x)37{{\sin \left(7\,x\right)-{{\sin ^3\left(7\,x\right)}\over{3}} }\over{7}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
Plot the graph f(x)
The answer [src] 
     3            
  sin (7)   sin(7)
- ------- + ------
     21       7
−sin⁡37−3 sin⁡721-{{\sin ^37-3\,\sin 7}\over{21}} 
=
= 
     3            
  sin (7)   sin(7)
- ------- + ------
     21       7
−sin⁡3(7)21+sin⁡(7)7- \frac{\sin^{3}{\left(7 \right)}}{21} + \frac{\sin{\left(7 \right)}}{7}
Упростить
Numerical answer [src] 
0.0803516074643471
0.0803516074643471
The graph
Integral of cos^37x dx

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

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