Integral of xsin-1xdx dx

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

Integral(x*sin(x) - x, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = 1$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \cos{\left(x \right)}\right)\, dx = - \int \cos{\left(x \right)}\, dx$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  So, the result is: $- \sin{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- x\right)\, dx = - \int x\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  So, the result is: $- \frac{x^{2}}{2}$
The result is: $- \frac{x^{2}}{2} - x \cos{\left(x \right)} + \sin{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-1/2 - cos(1) + sin(1)

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

=

-1/2 - cos(1) + sin(1)

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

-1/2 - cos(1) + sin(1)

Numerical answer [src]

-0.198831321060243

-0.198831321060243

The graph

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Identical expressions

Similar expressions

Integral of xsin-1xdx dx

Limits of integration:

The graph:

Piecewise:

The solution