Mister Exam

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xcos(3x^2-2)

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

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

The solution

You have entered [src]

  1                   
  /                   
 |                    
 |       /   2    \   
 |  x*cos\3*x  + 2/ dx
 |                    
/                     
0

$$\int\limits_{0}^{1} x \cos{\left(3 x^{2} + 2 \right)}\, dx$$

Integral(x*cos(3*x^2 + 2), (x, 0, 1))

Detail solution

Let $u = 3 x^{2} + 2$ .

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

$\int \frac{\cos{\left(u \right)}}{6}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{6}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
  So, the result is: $\frac{\sin{\left(u \right)}}{6}$
Now substitute $u$ back in:

$\frac{\sin{\left(3 x^{2} + 2 \right)}}{6}$
Now simplify:

$\frac{\sin{\left(3 x^{2} + 2 \right)}}{6}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                      
 |                             /   2    \
 |      /   2    \          sin\3*x  + 2/
 | x*cos\3*x  + 2/ dx = C + -------------
 |                                6      
/

$$\int x \cos{\left(3 x^{2} + 2 \right)}\, dx = C + \frac{\sin{\left(3 x^{2} + 2 \right)}}{6}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  sin(2)   sin(5)
- ------ + ------
    6        6

$$\frac{\sin{\left(5 \right)}}{6} - \frac{\sin{\left(2 \right)}}{6}$$

  sin(2)   sin(5)
- ------ + ------
    6        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

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution