Integral of xcosx+sinx dx

The solution

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

Integral(x*cos(x) + sin(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 $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = 1$ .
  
  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. The integral of sine is negative cosine:
  
  $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
1. The integral of sine is negative cosine:
  
  $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
The result is: $x \sin{\left(x \right)}$
Add the constant of integration:

$x \sin{\left(x \right)}+ \mathrm{constant}$

The answer is:

$x \sin{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

sin(1)

$$\sin{\left(1 \right)}$$

sin(1)

$$\sin{\left(1 \right)}$$

sin(1)

Numerical answer [src]

0.841470984807897

0.841470984807897

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

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Identical expressions

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Integral of xcosx+sinx dx

Limits of integration:

The graph:

Piecewise:

The solution