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Solving integrals/ 6cos^2x*sinx

Integral of 6cos^2x*sinx dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
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∫π2π6sin⁡(x)6cos⁡2(x) dx\int\limits_{\frac{\pi}{2}}^{\frac{\pi}{6}} \sin{\left(x \right)} 6 \cos^{2}{\left(x \right)}\, dx 
Integral((6*cos(x)^2)*sin(x), (x, pi/2, pi/6))
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 −6du- 6 du:

    ∫(−6u2) du\int \left(- 6 u^{2}\right)\, du

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

      ∫u2 du=−6∫u2 du\int u^{2}\, du = - 6 \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: −2u3- 2 u^{3}

    Now substitute uu back in:

    −2cos⁡3(x)- 2 \cos^{3}{\left(x \right)}

  2. Add the constant of integration:

    −2cos⁡3(x)+constant- 2 \cos^{3}{\left(x \right)}+ \mathrm{constant}

The answer is:

−2cos⁡3(x)+constant- 2 \cos^{3}{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                   
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 |      2                         3   
 | 6*cos (x)*sin(x) dx = C - 2*cos (x)
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∫sin⁡(x)6cos⁡2(x) dx=C−2cos⁡3(x)\int \sin{\left(x \right)} 6 \cos^{2}{\left(x \right)}\, dx = C - 2 \cos^{3}{\left(x \right)}
Simplify
The graph
0.600.700.800.901.001.101.201.301.401.505-5
Plot the graph f(x)
The answer [src] 
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-3*\/ 3 
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   4
−334- \frac{3 \sqrt{3}}{4} 
=
= 
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-3*\/ 3 
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   4
−334- \frac{3 \sqrt{3}}{4} 
-3*sqrt(3)/4
Numerical answer [src] 
-1.29903810567666
-1.29903810567666

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

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