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2sin(x)cos(5x)cos(x)dx
Solving integrals/ 2sin(x)cos(5x)cos(x)dx

Integral of 2sin(x)cos(5x)cos(x)dx dx

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∫012sin⁡(x)cos⁡(5x)cos⁡(x)1 dx\int\limits_{0}^{1} 2 \sin{\left(x \right)} \cos{\left(5 x \right)} \cos{\left(x \right)} 1\, dx 
Integral(2*sin(x)*cos(5*x)*cos(x)*1, (x, 0, 1))
Detail solution

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

    ∫2sin⁡(x)cos⁡(5x)cos⁡(x)1 dx=2∫sin⁡(x)cos⁡(x)cos⁡(5x) dx\int 2 \sin{\left(x \right)} \cos{\left(5 x \right)} \cos{\left(x \right)} 1\, dx = 2 \int \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(5 x \right)}\, dx

    1. Rewrite the integrand:

      sin⁡(x)cos⁡(x)cos⁡(5x)=16sin⁡(x)cos⁡6(x)−20sin⁡(x)cos⁡4(x)+5sin⁡(x)cos⁡2(x)\sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(5 x \right)} = 16 \sin{\left(x \right)} \cos^{6}{\left(x \right)} - 20 \sin{\left(x \right)} \cos^{4}{\left(x \right)} + 5 \sin{\left(x \right)} \cos^{2}{\left(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:

        ∫16sin⁡(x)cos⁡6(x) dx=16∫sin⁡(x)cos⁡6(x) dx\int 16 \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx = 16 \int \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫u6 du\int u^{6}\, du

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

            ∫(−u6) du=−∫u6 du\int \left(- u^{6}\right)\, du = - \int u^{6}\, du

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

              ∫u6 du=u77\int u^{6}\, du = \frac{u^{7}}{7}

            So, the result is: −u77- \frac{u^{7}}{7}

          Now substitute uu back in:

          −cos⁡7(x)7- \frac{\cos^{7}{\left(x \right)}}{7}

        So, the result is: −16cos⁡7(x)7- \frac{16 \cos^{7}{\left(x \right)}}{7}

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

        ∫(−20sin⁡(x)cos⁡4(x)) dx=−20∫sin⁡(x)cos⁡4(x) dx\int \left(- 20 \sin{\left(x \right)} \cos^{4}{\left(x \right)}\right)\, dx = - 20 \int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫u4 du\int u^{4}\, du

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

            ∫(−u4) du=−∫u4 du\int \left(- u^{4}\right)\, du = - \int u^{4}\, du

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

              ∫u4 du=u55\int u^{4}\, du = \frac{u^{5}}{5}

            So, the result is: −u55- \frac{u^{5}}{5}

          Now substitute uu back in:

          −cos⁡5(x)5- \frac{\cos^{5}{\left(x \right)}}{5}

        So, the result is: 4cos⁡5(x)4 \cos^{5}{\left(x \right)}

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

        ∫5sin⁡(x)cos⁡2(x) dx=5∫sin⁡(x)cos⁡2(x) dx\int 5 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = 5 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx

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

          Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

          ∫u2 du\int u^{2}\, du

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

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

            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: −u33- \frac{u^{3}}{3}

          Now substitute uu back in:

          −cos⁡3(x)3- \frac{\cos^{3}{\left(x \right)}}{3}

        So, the result is: −5cos⁡3(x)3- \frac{5 \cos^{3}{\left(x \right)}}{3}

      The result is: −16cos⁡7(x)7+4cos⁡5(x)−5cos⁡3(x)3- \frac{16 \cos^{7}{\left(x \right)}}{7} + 4 \cos^{5}{\left(x \right)} - \frac{5 \cos^{3}{\left(x \right)}}{3}

    So, the result is: −32cos⁡7(x)7+8cos⁡5(x)−10cos⁡3(x)3- \frac{32 \cos^{7}{\left(x \right)}}{7} + 8 \cos^{5}{\left(x \right)} - \frac{10 \cos^{3}{\left(x \right)}}{3}

  2. Now simplify:

    2(−48cos⁡4(x)+84cos⁡2(x)−35)cos⁡3(x)21\frac{2 \left(- 48 \cos^{4}{\left(x \right)} + 84 \cos^{2}{\left(x \right)} - 35\right) \cos^{3}{\left(x \right)}}{21}

  3. Add the constant of integration:

    2(−48cos⁡4(x)+84cos⁡2(x)−35)cos⁡3(x)21+constant\frac{2 \left(- 48 \cos^{4}{\left(x \right)} + 84 \cos^{2}{\left(x \right)} - 35\right) \cos^{3}{\left(x \right)}}{21}+ \mathrm{constant}

The answer is:

2(−48cos⁡4(x)+84cos⁡2(x)−35)cos⁡3(x)21+constant\frac{2 \left(- 48 \cos^{4}{\left(x \right)} + 84 \cos^{2}{\left(x \right)} - 35\right) \cos^{3}{\left(x \right)}}{21}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                      7            3   
 |                                          5      32*cos (x)   10*cos (x)
 | 2*sin(x)*cos(5*x)*cos(x)*1 dx = C + 8*cos (x) - ---------- - ----------
 |                                                     7            3     
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cos⁡(3 x)3−cos⁡(7 x)72{{{{\cos \left(3\,x\right)}\over{3}}-{{\cos \left(7\,x\right) }\over{7}}}\over{2}}
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The graph
0.001.000.100.200.300.400.500.600.700.800.901-1
Plot the graph f(x)
The answer [src] 
            2                  2                                    
  2    2*sin (1)*cos(5)   2*cos (1)*cos(5)   10*cos(1)*sin(1)*sin(5)
- -- - ---------------- + ---------------- + -----------------------
  21          21                 21                     21
2 (−3 cos⁡7−7 cos⁡384−121)2\,\left(-{{3\,\cos 7-7\,\cos 3}\over{84}}-{{1}\over{21}}\right) 
=
= 
            2                  2                                    
  2    2*sin (1)*cos(5)   2*cos (1)*cos(5)   10*cos(1)*sin(1)*sin(5)
- -- - ---------------- + ---------------- + -----------------------
  21          21                 21                     21
10sin⁡(1)sin⁡(5)cos⁡(1)21−221−2sin⁡2(1)cos⁡(5)21+2cos⁡2(1)cos⁡(5)21\frac{10 \sin{\left(1 \right)} \sin{\left(5 \right)} \cos{\left(1 \right)}}{21} - \frac{2}{21} - \frac{2 \sin^{2}{\left(1 \right)} \cos{\left(5 \right)}}{21} + \frac{2 \cos^{2}{\left(1 \right)} \cos{\left(5 \right)}}{21}
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Numerical answer [src] 
-0.314087005696025
-0.314087005696025
The graph
Integral of 2sin(x)cos(5x)cos(x)dx dx

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

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