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Solving integrals/ xcos(x+3)

Integral of xcos(x+3) dx

The solution

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 |  x*cos(x + 3) dx
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0

$$\int\limits_{0}^{1} x \cos{\left(x + 3 \right)}\, dx$$

Integral(x*cos(x + 3), (x, 0, 1))

Detail solution

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 $\operatorname{dv}{\left(x \right)} = \cos{\left(x + 3 \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 1$ .

To find $v{\left(x \right)}$ :
1. Let $u = x + 3$ .
  
  Then let $du = dx$ and substitute $du$ :
  
  $\int \cos{\left(u \right)}\, du$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
  Now substitute $u$ back in:
  
  $\sin{\left(x + 3 \right)}$
Now evaluate the sub-integral.
Let $u = x + 3$ .

Then let $du = dx$ and substitute $du$ :

$\int \sin{\left(u \right)}\, du$
1. The integral of sine is negative cosine:
  
  $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
Now substitute $u$ back in:

$- \cos{\left(x + 3 \right)}$
Add the constant of integration:

$x \sin{\left(x + 3 \right)} + \cos{\left(x + 3 \right)}+ \mathrm{constant}$

The answer is:

$x \sin{\left(x + 3 \right)} + \cos{\left(x + 3 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                               
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 | x*cos(x + 3) dx = C + x*sin(3 + x) + cos(3 + x)
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$$\int x \cos{\left(x + 3 \right)}\, dx = C + x \sin{\left(x + 3 \right)} + \cos{\left(x + 3 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-cos(3) + cos(4) + sin(4)

$$\sin{\left(4 \right)} + \cos{\left(4 \right)} - \cos{\left(3 \right)}$$

-cos(3) + cos(4) + sin(4)

$$\sin{\left(4 \right)} + \cos{\left(4 \right)} - \cos{\left(3 \right)}$$

-cos(3) + cos(4) + sin(4)

Numerical answer [src]

-0.420453619571095

-0.420453619571095

The graph

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

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Identical expressions

Similar expressions

Integral of xcos(x+3) dx

Limits of integration:

The graph:

Piecewise:

The solution