Integral of xsen(2x²) dx

The solution

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  1               
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 |       /   2\   
 |  x*sin\2*x / dx
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0

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

Integral(x*sin(2*x^2), (x, 0, 1))

Detail solution

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

Then let $du = 4 x dx$ and substitute $\frac{du}{4}$ :

$\int \frac{\sin{\left(u \right)}}{4}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $- \frac{\cos{\left(u \right)}}{4}$
Now substitute $u$ back in:

$- \frac{\cos{\left(2 x^{2} \right)}}{4}$
Add the constant of integration:

$- \frac{\cos{\left(2 x^{2} \right)}}{4}+ \mathrm{constant}$

The answer is:

$- \frac{\cos{\left(2 x^{2} \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                              
 |                         /   2\
 |      /   2\          cos\2*x /
 | x*sin\2*x / dx = C - ---------
 |                          4    
/

$$\int x \sin{\left(2 x^{2} \right)}\, dx = C - \frac{\cos{\left(2 x^{2} \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   cos(2)
- - ------
4     4

$$\frac{1}{4} - \frac{\cos{\left(2 \right)}}{4}$$

1   cos(2)
- - ------
4     4

$$\frac{1}{4} - \frac{\cos{\left(2 \right)}}{4}$$

1/4 - cos(2)/4

Numerical answer [src]

0.354036709136786

0.354036709136786

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

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Identical expressions

Integral of xsen(2x²) dx

Limits of integration:

The graph:

Piecewise:

The solution