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Solving integrals/ x^2*sqrt(2)cos(x)

Integral of x^2*sqrt(2)cos(x) dx

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The solution

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∫0π42x2cos⁡(x) dx\int\limits_{0}^{\frac{\pi}{4}} \sqrt{2} x^{2} \cos{\left(x \right)}\, dx 
Integral((x^2*sqrt(2))*cos(x), (x, 0, pi/4))
Detail solution

  1. Use integration by parts:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=2x2u{\left(x \right)} = \sqrt{2} x^{2} and let dv⁡(x)=cos⁡(x)\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}.

    Then du⁡(x)=22x\operatorname{du}{\left(x \right)} = 2 \sqrt{2} x.

    To find v(x)v{\left(x \right)}:

    1. The integral of cosine is sine:

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

    Now evaluate the sub-integral.

  2. Use integration by parts:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=22xu{\left(x \right)} = 2 \sqrt{2} x and let dv⁡(x)=sin⁡(x)\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}.

    Then du⁡(x)=22\operatorname{du}{\left(x \right)} = 2 \sqrt{2}.

    To find v(x)v{\left(x \right)}:

    1. The integral of sine is negative cosine:

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

    Now evaluate the sub-integral.

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

    ∫(−22cos⁡(x)) dx=−22∫cos⁡(x) dx\int \left(- 2 \sqrt{2} \cos{\left(x \right)}\right)\, dx = - 2 \sqrt{2} \int \cos{\left(x \right)}\, dx

    1. The integral of cosine is sine:

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

    So, the result is: −22sin⁡(x)- 2 \sqrt{2} \sin{\left(x \right)}

  4. Now simplify:

    2(x2sin⁡(x)+2xcos⁡(x)−2sin⁡(x))\sqrt{2} \left(x^{2} \sin{\left(x \right)} + 2 x \cos{\left(x \right)} - 2 \sin{\left(x \right)}\right)

  5. Add the constant of integration:

    2(x2sin⁡(x)+2xcos⁡(x)−2sin⁡(x))+constant\sqrt{2} \left(x^{2} \sin{\left(x \right)} + 2 x \cos{\left(x \right)} - 2 \sin{\left(x \right)}\right)+ \mathrm{constant}

The answer is:

2(x2sin⁡(x)+2xcos⁡(x)−2sin⁡(x))+constant\sqrt{2} \left(x^{2} \sin{\left(x \right)} + 2 x \cos{\left(x \right)} - 2 \sin{\left(x \right)}\right)+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                            
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 |  2   ___                     ___            ___  2                ___       
 | x *\/ 2 *cos(x) dx = C - 2*\/ 2 *sin(x) + \/ 2 *x *sin(x) + 2*x*\/ 2 *cos(x)
 |                                                                             
/
∫2x2cos⁡(x) dx=C+2x2sin⁡(x)+22xcos⁡(x)−22sin⁡(x)\int \sqrt{2} x^{2} \cos{\left(x \right)}\, dx = C + \sqrt{2} x^{2} \sin{\left(x \right)} + 2 \sqrt{2} x \cos{\left(x \right)} - 2 \sqrt{2} \sin{\left(x \right)}
Simplify
The graph
0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.01.0
Plot the graph f(x)
The answer [src] 
      /               ___     ___   2\
  ___ |    ___   pi*\/ 2    \/ 2 *pi |
\/ 2 *|- \/ 2  + -------- + ---------|
      \             4           32   /
2(−2+2π232+2π4)\sqrt{2} \left(- \sqrt{2} + \frac{\sqrt{2} \pi^{2}}{32} + \frac{\sqrt{2} \pi}{4}\right) 
=
= 
      /               ___     ___   2\
  ___ |    ___   pi*\/ 2    \/ 2 *pi |
\/ 2 *|- \/ 2  + -------- + ---------|
      \             4           32   /
2(−2+2π232+2π4)\sqrt{2} \left(- \sqrt{2} + \frac{\sqrt{2} \pi^{2}}{32} + \frac{\sqrt{2} \pi}{4}\right) 
sqrt(2)*(-sqrt(2) + pi*sqrt(2)/4 + sqrt(2)*pi^2/32)
Numerical answer [src] 
0.187646601862982
0.187646601862982

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

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