Mister Exam

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Integral of x^2*cos(x)+2 dx

The solution

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0

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

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

Numerical answer [src]

2.23913362692838

2.23913362692838

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

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Integral of x^2*cos(x)+2 dx

Limits of integration:

The graph:

Piecewise:

The solution