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

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

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∫01sin⁡(x)cos⁡(2x) dx\int\limits_{0}^{1} \sin{\left(x \right)} \cos{\left(2 x \right)}\, dx 
Integral(sin(x)*cos(2*x), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    sin⁡(x)cos⁡(2x)=2sin⁡(x)cos⁡2(x)−sin⁡(x)\sin{\left(x \right)} \cos{\left(2 x \right)} = 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)} - \sin{\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:

      ∫2sin⁡(x)cos⁡2(x) dx=2∫sin⁡(x)cos⁡2(x) dx\int 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = 2 \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: −2cos⁡3(x)3- \frac{2 \cos^{3}{\left(x \right)}}{3}

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

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

      1. The integral of sine is negative cosine:

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

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

    The result is: −2cos⁡3(x)3+cos⁡(x)- \frac{2 \cos^{3}{\left(x \right)}}{3} + \cos{\left(x \right)}

  3. Now simplify:

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

  4. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                              3            
 |                          2*cos (x)         
 | sin(x)*cos(2*x) dx = C - --------- + cos(x)
 |                              3             
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cos⁡x2−cos⁡(3 x)6{{\cos x}\over{2}}-{{\cos \left(3\,x\right)}\over{6}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901.0-1.0
Plot the graph f(x)
The answer [src] 
  1   cos(1)*cos(2)   2*sin(1)*sin(2)
- - + ------------- + ---------------
  3         3                3
−cos⁡3−3 cos⁡16−13-{{\cos 3-3\,\cos 1}\over{6}}-{{1}\over{3}} 
=
= 
  1   cos(1)*cos(2)   2*sin(1)*sin(2)
- - + ------------- + ---------------
  3         3                3
−13+cos⁡(1)cos⁡(2)3+2sin⁡(1)sin⁡(2)3- \frac{1}{3} + \frac{\cos{\left(1 \right)} \cos{\left(2 \right)}}{3} + \frac{2 \sin{\left(1 \right)} \sin{\left(2 \right)}}{3}
Simplify
Numerical answer [src] 
0.101816569034144
0.101816569034144
The graph
Integral of sin(x)cos(2x) dx

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

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