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xcos(3x^2+2)
Solving integrals/ xcos(3x^2+2)

Integral of xcos(3x^2+2) dx

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The solution

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∫01xcos⁡(3x2+2) dx\int\limits_{0}^{1} x \cos{\left(3 x^{2} + 2 \right)}\, dx 
Integral(x*cos(3*x^2 + 2), (x, 0, 1))
Detail solution

  1. Let u=3x2+2u = 3 x^{2} + 2.

    Then let du=6xdxdu = 6 x dx and substitute du6\frac{du}{6}:

    ∫cos⁡(u)6 du\int \frac{\cos{\left(u \right)}}{6}\, du

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

      ∫cos⁡(u) du=∫cos⁡(u) du6\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{6}

      1. The integral of cosine is sine:

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

      So, the result is: sin⁡(u)6\frac{\sin{\left(u \right)}}{6}

    Now substitute uu back in:

    sin⁡(3x2+2)6\frac{\sin{\left(3 x^{2} + 2 \right)}}{6}

  2. Now simplify:

    sin⁡(3x2+2)6\frac{\sin{\left(3 x^{2} + 2 \right)}}{6}

  3. Add the constant of integration:

    sin⁡(3x2+2)6+constant\frac{\sin{\left(3 x^{2} + 2 \right)}}{6}+ \mathrm{constant}

The answer is:

sin⁡(3x2+2)6+constant\frac{\sin{\left(3 x^{2} + 2 \right)}}{6}+ \mathrm{constant}

The answer (Indefinite) [src] 
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 | x*cos\3*x  + 2/ dx = C + -------------
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∫xcos⁡(3x2+2) dx=C+sin⁡(3x2+2)6\int x \cos{\left(3 x^{2} + 2 \right)}\, dx = C + \frac{\sin{\left(3 x^{2} + 2 \right)}}{6}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901-1
Plot the graph f(x)
The answer [src] 
  sin(2)   sin(5)
- ------ + ------
    6        6
sin⁡(5)6−sin⁡(2)6\frac{\sin{\left(5 \right)}}{6} - \frac{\sin{\left(2 \right)}}{6} 
=
= 
  sin(2)   sin(5)
- ------ + ------
    6        6
sin⁡(5)6−sin⁡(2)6\frac{\sin{\left(5 \right)}}{6} - \frac{\sin{\left(2 \right)}}{6} 
-sin(2)/6 + sin(5)/6
Numerical answer [src] 
-0.31137028358147
-0.31137028358147
The graph
Integral of xcos(3x^2+2) dx

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

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