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Solving integrals/ sqrt(2sin3x)cos3x

Integral of sqrt(2sin3x)cos3x 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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 |  \/ 2*sin(3*x) *cos(3*x) dx
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∫012sin⁡(3x)cos⁡(3x) dx\int\limits_{0}^{1} \sqrt{2 \sin{\left(3 x \right)}} \cos{\left(3 x \right)}\, dx 
Integral(sqrt(2*sin(3*x))*cos(3*x), (x, 0, 1))
Detail solution

  1. Let u=3xu = 3 x.

    Then let du=3dxdu = 3 dx and substitute 2du3\frac{\sqrt{2} du}{3}:

    ∫2sin⁡(u)cos⁡(u)3 du\int \frac{\sqrt{2} \sqrt{\sin{\left(u \right)}} \cos{\left(u \right)}}{3}\, du

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

      ∫sin⁡(u)cos⁡(u) du=2∫sin⁡(u)cos⁡(u) du3\int \sqrt{\sin{\left(u \right)}} \cos{\left(u \right)}\, du = \frac{\sqrt{2} \int \sqrt{\sin{\left(u \right)}} \cos{\left(u \right)}\, du}{3}

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

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

        ∫u du\int \sqrt{u}\, du

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

          ∫u du=2u323\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}

        Now substitute uu back in:

        2sin⁡32(u)3\frac{2 \sin^{\frac{3}{2}}{\left(u \right)}}{3}

      So, the result is: 22sin⁡32(u)9\frac{2 \sqrt{2} \sin^{\frac{3}{2}}{\left(u \right)}}{9}

    Now substitute uu back in:

    22sin⁡32(3x)9\frac{2 \sqrt{2} \sin^{\frac{3}{2}}{\left(3 x \right)}}{9}

  2. Add the constant of integration:

    22sin⁡32(3x)9+constant\frac{2 \sqrt{2} \sin^{\frac{3}{2}}{\left(3 x \right)}}{9}+ \mathrm{constant}

The answer is:

22sin⁡32(3x)9+constant\frac{2 \sqrt{2} \sin^{\frac{3}{2}}{\left(3 x \right)}}{9}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                    
 |                                      ___    3/2     
 |   ____________                   2*\/ 2 *sin   (3*x)
 | \/ 2*sin(3*x) *cos(3*x) dx = C + -------------------
 |                                           9         
/
∫2sin⁡(3x)cos⁡(3x) dx=C+22sin⁡32(3x)9\int \sqrt{2 \sin{\left(3 x \right)}} \cos{\left(3 x \right)}\, dx = C + \frac{2 \sqrt{2} \sin^{\frac{3}{2}}{\left(3 x \right)}}{9}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
Plot the graph f(x)
The answer [src] 
    ___    3/2   
2*\/ 2 *sin   (3)
-----------------
        9
22sin⁡32(3)9\frac{2 \sqrt{2} \sin^{\frac{3}{2}}{\left(3 \right)}}{9} 
=
= 
    ___    3/2   
2*\/ 2 *sin   (3)
-----------------
        9
22sin⁡32(3)9\frac{2 \sqrt{2} \sin^{\frac{3}{2}}{\left(3 \right)}}{9} 
2*sqrt(2)*sin(3)^(3/2)/9
Numerical answer [src] 
0.0166603980343461
0.0166603980343461

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

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