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x*sin(x²)
Solving integrals/ x*sin(x²)

Integral of x*sin(x²) dx

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The solution

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∫01xsin⁡(x2) dx\int\limits_{0}^{1} x \sin{\left(x^{2} \right)}\, dx 
Integral(x*sin(x^2), (x, 0, 1))
Detail solution

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

    Then let du=2xdxdu = 2 x dx and substitute du2\frac{du}{2}:

    ∫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)2 du=∫sin⁡(u) du2\int \frac{\sin{\left(u \right)}}{2}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}

      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)2- \frac{\cos{\left(u \right)}}{2}

    Now substitute uu back in:

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

  2. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                          
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 |      / 2\          cos\x /
 | x*sin\x / dx = C - -------
 |                       2   
/
−cos⁡x22-{{\cos x^2}\over{2}}
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(1)
- - ------
2     2
12−cos⁡12{{1}\over{2}}-{{\cos 1}\over{2}} 
=
= 
1   cos(1)
- - ------
2     2
−cos⁡(1)2+12- \frac{\cos{\left(1 \right)}}{2} + \frac{1}{2}
Simplify
Numerical answer [src] 
0.22984884706593
0.22984884706593
The graph
Integral of x*sin(x²) dx

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

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