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Solving integrals/ xsen(2x²)

Integral of xsen(2x²) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1               
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 |  x*sin\2*x / dx
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∫01xsin⁡(2x2) dx\int\limits_{0}^{1} x \sin{\left(2 x^{2} \right)}\, dx 
Integral(x*sin(2*x^2), (x, 0, 1))
Detail solution

  1. Let u=2x2u = 2 x^{2}.

    Then let du=4xdxdu = 4 x dx and substitute du4\frac{du}{4}:

    ∫sin⁡(u)4 du\int \frac{\sin{\left(u \right)}}{4}\, du

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

      ∫sin⁡(u) du=∫sin⁡(u) du4\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}

      1. The integral of sine is negative cosine:

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

      So, the result is: −cos⁡(u)4- \frac{\cos{\left(u \right)}}{4}

    Now substitute uu back in:

    −cos⁡(2x2)4- \frac{\cos{\left(2 x^{2} \right)}}{4}

  2. Add the constant of integration:

    −cos⁡(2x2)4+constant- \frac{\cos{\left(2 x^{2} \right)}}{4}+ \mathrm{constant}

The answer is:

−cos⁡(2x2)4+constant- \frac{\cos{\left(2 x^{2} \right)}}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                              
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 |      /   2\          cos\2*x /
 | x*sin\2*x / dx = C - ---------
 |                          4    
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∫xsin⁡(2x2) dx=C−cos⁡(2x2)4\int x \sin{\left(2 x^{2} \right)}\, dx = C - \frac{\cos{\left(2 x^{2} \right)}}{4}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901-1
Plot the graph f(x)
The answer [src] 
1   cos(2)
- - ------
4     4
14−cos⁡(2)4\frac{1}{4} - \frac{\cos{\left(2 \right)}}{4} 
=
= 
1   cos(2)
- - ------
4     4
14−cos⁡(2)4\frac{1}{4} - \frac{\cos{\left(2 \right)}}{4} 
1/4 - cos(2)/4
Numerical answer [src] 
0.354036709136786
0.354036709136786

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

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