Integral of 3xsinx dx

The solution

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 |  3*x*sin(x) dx
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$$\int\limits_{\frac{x}{2}}^{x} 3 x \sin{\left(x \right)}\, dx$$

Integral((3*x)*sin(x), (x, x/2, x))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = 3 x$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 3$ .

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.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- 3 \cos{\left(x \right)}\right)\, dx = - 3 \int \cos{\left(x \right)}\, dx$
1. The integral of cosine is sine:
  
  $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
So, the result is: $- 3 \sin{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

Simplify

The answer [src]

                                            /x\
                                     3*x*cos|-|
       /x\                                  \2/
- 3*sin|-| + 3*sin(x) - 3*x*cos(x) + ----------
       \2/                               2

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

                                            /x\
                                     3*x*cos|-|
       /x\                                  \2/
- 3*sin|-| + 3*sin(x) - 3*x*cos(x) + ----------
       \2/                               2

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

-3*sin(x/2) + 3*sin(x) - 3*x*cos(x) + 3*x*cos(x/2)/2

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

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Integral of 3xsinx dx

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The solution