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Solving integrals/ sin(x^2)x

Integral of sin(x^2)x dx

The solution

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

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

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

Detail solution

Let $u = x^{2}$ .

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

$\int \frac{\sin{\left(u \right)}}{2}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $- \frac{\cos{\left(u \right)}}{2}$
Now substitute $u$ back in:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

1/2 - cos(1)/2

Numerical answer [src]

0.22984884706593

0.22984884706593

The graph

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

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Identical expressions

Integral of sin(x^2)x dx

Limits of integration:

The graph:

Piecewise:

The solution