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xsin(2x-3)dx

Solving integrals/ xsin(2x+3)dx

Integral of xsin(2x+3)dx dx

The solution

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 |  x*sin(2*x + 3) dx
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0

$$\int\limits_{0}^{1} x \sin{\left(2 x + 3 \right)}\, dx$$

Integral(x*sin(2*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 $\operatorname{dv}{\left(x \right)} = \sin{\left(2 x + 3 \right)}$ .

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

To find $v{\left(x \right)}$ :
1. Let $u = 2 x + 3$ .
  
  Then let $du = 2 dx$ 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:
  
  $- \frac{\cos{\left(2 x + 3 \right)}}{2}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{\cos{\left(2 x + 3 \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(2 x + 3 \right)}\, dx}{2}$
1. Let $u = 2 x + 3$ .
  
  Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{\cos{\left(u \right)}}{2}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{2}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(2 x + 3 \right)}}{2}$
So, the result is: $- \frac{\sin{\left(2 x + 3 \right)}}{4}$
Add the constant of integration:

$- \frac{x \cos{\left(2 x + 3 \right)}}{2} + \frac{\sin{\left(2 x + 3 \right)}}{4}+ \mathrm{constant}$

The answer is:

$- \frac{x \cos{\left(2 x + 3 \right)}}{2} + \frac{\sin{\left(2 x + 3 \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                     
 |                         sin(3 + 2*x)   x*cos(3 + 2*x)
 | x*sin(2*x + 3) dx = C + ------------ - --------------
 |                              4               2       
/

$$\int x \sin{\left(2 x + 3 \right)}\, dx = C - \frac{x \cos{\left(2 x + 3 \right)}}{2} + \frac{\sin{\left(2 x + 3 \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  cos(5)   sin(3)   sin(5)
- ------ - ------ + ------
    2        4        4

$$\frac{\sin{\left(5 \right)}}{4} - \frac{\cos{\left(5 \right)}}{2} - \frac{\sin{\left(3 \right)}}{4}$$

  cos(5)   sin(3)   sin(5)
- ------ - ------ + ------
    2        4        4

$$\frac{\sin{\left(5 \right)}}{4} - \frac{\cos{\left(5 \right)}}{2} - \frac{\sin{\left(3 \right)}}{4}$$

-cos(5)/2 - sin(3)/4 + sin(5)/4

Numerical answer [src]

-0.416842163412365

-0.416842163412365

The graph

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

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Identical expressions

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Integral of xsin(2x+3)dx dx

Limits of integration:

The graph:

Piecewise:

The solution