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sinx*cos^6x*dx
Solving integrals/ sinx*cos^6x*dx

Integral of sinx*cos^6x*dx dx

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The solution

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∫01sin⁡(x)cos⁡6(x)1 dx\int\limits_{0}^{1} \sin{\left(x \right)} \cos^{6}{\left(x \right)} 1\, dx 
Integral(sin(x)*cos(x)^6*1, (x, 0, 1))
Detail solution

  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}

  2. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                 
 |                              7   
 |           6               cos (x)
 | sin(x)*cos (x)*1 dx = C - -------
 |                              7   
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∫sin⁡(x)cos⁡6(x)1 dx=C−cos⁡7(x)7\int \sin{\left(x \right)} \cos^{6}{\left(x \right)} 1\, dx = C - \frac{\cos^{7}{\left(x \right)}}{7}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.5-0.5
Plot the graph f(x)
The answer [src] 
       7   
1   cos (1)
- - -------
7      7
17−cos⁡7(1)7\frac{1}{7} - \frac{\cos^{7}{\left(1 \right)}}{7} 
=
= 
       7   
1   cos (1)
- - -------
7      7
17−cos⁡7(1)7\frac{1}{7} - \frac{\cos^{7}{\left(1 \right)}}{7}
Упростить
Numerical answer [src] 
0.140936884309461
0.140936884309461
The graph
Integral of sinx*cos^6x*dx dx

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

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