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Similar expressions
x^2cos(3x^3-5)

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

The solution

You have entered [src]

 oo                    
  /                    
 |                     
 |   2    /   3    \   
 |  x *cos\3*x  + 5/ dx
 |                     
/                      
1

$$\int\limits_{1}^{\infty} x^{2} \cos{\left(3 x^{3} + 5 \right)}\, dx$$

Integral(x^2*cos(3*x^3 + 5), (x, 1, oo))

Detail solution

Let $u = 3 x^{3} + 5$ .

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

$\int \frac{\cos{\left(u \right)}}{81}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\cos{\left(u \right)}}{9}\, du = \frac{\int \cos{\left(u \right)}\, du}{9}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
  So, the result is: $\frac{\sin{\left(u \right)}}{9}$
Now substitute $u$ back in:

$\frac{\sin{\left(3 x^{3} + 5 \right)}}{9}$
Now simplify:

$\frac{\sin{\left(3 x^{3} + 5 \right)}}{9}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                       
 |                              /   3    \
 |  2    /   3    \          sin\3*x  + 5/
 | x *cos\3*x  + 5/ dx = C + -------------
 |                                 9      
/

$$\int x^{2} \cos{\left(3 x^{3} + 5 \right)}\, dx = C + \frac{\sin{\left(3 x^{3} + 5 \right)}}{9}$$

Simplify

The answer [src]

   1   sin(8)  1   sin(8) 
<- - - ------, - - ------>
   9     9     9     9

$$\left\langle - \frac{1}{9} - \frac{\sin{\left(8 \right)}}{9}, \frac{1}{9} - \frac{\sin{\left(8 \right)}}{9}\right\rangle$$

   1   sin(8)  1   sin(8) 
<- - - ------, - - ------>
   9     9     9     9

$$\left\langle - \frac{1}{9} - \frac{\sin{\left(8 \right)}}{9}, \frac{1}{9} - \frac{\sin{\left(8 \right)}}{9}\right\rangle$$

Упростить

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

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

Limits of integration:

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Piecewise:

The solution