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sin^6*x*cosxdx
Solving integrals/ sin^6*x*cosxdx

Integral of sin^6*x*cosxdx dx

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The solution

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

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

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

    ∫u6 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}

    Now substitute uu back in:

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

  2. Add the constant of integration:

    sin⁡7(x)7+constant\frac{\sin^{7}{\left(x \right)}}{7}+ \mathrm{constant}

The answer is:

sin⁡7(x)7+constant\frac{\sin^{7}{\left(x \right)}}{7}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                               
 |                            7   
 |    6                    sin (x)
 | sin (x)*cos(x) dx = C + -------
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∫sin⁡6(x)cos⁡(x) dx=C+sin⁡7(x)7\int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx = C + \frac{\sin^{7}{\left(x \right)}}{7}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.00.2
Plot the graph f(x)
The answer [src] 
   7   
sin (1)
-------
   7
sin⁡7(1)7\frac{\sin^{7}{\left(1 \right)}}{7} 
=
= 
   7   
sin (1)
-------
   7
sin⁡7(1)7\frac{\sin^{7}{\left(1 \right)}}{7} 
sin(1)^7/7
Numerical answer [src] 
0.0426752405751304
0.0426752405751304
The graph
Integral of sin^6*x*cosxdx dx

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

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