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Solving integrals/ x^2cos(3x^3+5)

Integral of x^2cos(3x^3+5) dx

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The solution

You have entered [src] 
 oo                    
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 |   2    /   3    \   
 |  x *cos\3*x  + 5/ dx
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∫1∞x2cos⁡(3x3+5) dx\int\limits_{1}^{\infty} x^{2} \cos{\left(3 x^{3} + 5 \right)}\, dx 
Integral(x^2*cos(3*x^3 + 5), (x, 1, oo))
Detail solution

  1. Let u=3x3+5u = 3 x^{3} + 5.

    Then let du=9x2dxdu = 9 x^{2} dx and substitute du9\frac{du}{9}:

    ∫cos⁡(u)81 du\int \frac{\cos{\left(u \right)}}{81}\, du

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

      ∫cos⁡(u)9 du=∫cos⁡(u) du9\int \frac{\cos{\left(u \right)}}{9}\, du = \frac{\int \cos{\left(u \right)}\, du}{9}

      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)9\frac{\sin{\left(u \right)}}{9}

    Now substitute uu back in:

    sin⁡(3x3+5)9\frac{\sin{\left(3 x^{3} + 5 \right)}}{9}

  2. Now simplify:

    sin⁡(3x3+5)9\frac{\sin{\left(3 x^{3} + 5 \right)}}{9}

  3. Add the constant of integration:

    sin⁡(3x3+5)9+constant\frac{\sin{\left(3 x^{3} + 5 \right)}}{9}+ \mathrm{constant}

The answer is:

sin⁡(3x3+5)9+constant\frac{\sin{\left(3 x^{3} + 5 \right)}}{9}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                       
 |                              /   3    \
 |  2    /   3    \          sin\3*x  + 5/
 | x *cos\3*x  + 5/ dx = C + -------------
 |                                 9      
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∫x2cos⁡(3x3+5) dx=C+sin⁡(3x3+5)9\int x^{2} \cos{\left(3 x^{3} + 5 \right)}\, dx = C + \frac{\sin{\left(3 x^{3} + 5 \right)}}{9}
Simplify
The answer [src] 
   1   sin(8)  1   sin(8) 
<- - - ------, - - ------>
   9     9     9     9
⟨−19−sin⁡(8)9,19−sin⁡(8)9⟩\left\langle - \frac{1}{9} - \frac{\sin{\left(8 \right)}}{9}, \frac{1}{9} - \frac{\sin{\left(8 \right)}}{9}\right\rangle 
=
= 
   1   sin(8)  1   sin(8) 
<- - - ------, - - ------>
   9     9     9     9
⟨−19−sin⁡(8)9,19−sin⁡(8)9⟩\left\langle - \frac{1}{9} - \frac{\sin{\left(8 \right)}}{9}, \frac{1}{9} - \frac{\sin{\left(8 \right)}}{9}\right\rangle
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    Use the examples entering the upper and lower limits of integration.

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