What is $\int x sin (\frac{x}{2} - π)$ $int xsin(x/2-pi)$ ?

Question

What is $\int x sin (\frac{x}{2} - π)$ $int xsin(x/2-pi)$ ?

1 Answer

Impact of this question

1583 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-04-20T03:01:57

$2 x cos (\frac{x}{2}) - 4 sin (\frac{x}{2}) + C$ $2xcos(x/2)-4sin(x/2)+C$

Explanation:

First, we can simplify $sin (\frac{x}{2} - π)$ $sin(x/2-pi)$ using the sine angle subtraction formula:

$sin (A - B) = sin A cos B - cos A sin B$ $sin(A-B)=sinAcosB-cosAsinB$

Thus,

$sin (\frac{x}{2} - π) = sin (\frac{x}{2}) cos (π) - cos (\frac{x}{2}) sin (π)$ $sin(x/2-pi)=sin(x/2)cos(pi)-cos(x/2)sin(pi)$

$= sin (\frac{x}{2}) (- 1) - cos (\frac{x}{2}) (0)$ $=sin(x/2)(-1)-cos(x/2)(0)$

$= - sin (\frac{x}{2})$ $=-sin(x/2)$

Thus the integral is slightly simplified to become

$= - \int x sin (\frac{x}{2}) d x$ $=-intxsin(x/2)dx$

First, set $t = \frac{x}{2}$ $t=x/2$ . This implies that $d t = \frac{1}{2} d x$ $dt=1/2dx$ and $x = 2 t$ $x=2t$ . We will want to multiply the interior of the integral by $\frac{1}{2}$ $1/2$ so that we have $\frac{1}{2} d x = t$ $1/2dx=t$ and balance this by multiplying the exterior by $2$ $2$ .

$= - 2 \int x sin (\frac{x}{2}) (\frac{1}{2}) d x$ $=-2intxsin(x/2)(1/2)dx$

Substitute in $x = 2 t$ $x=2t$ , $\frac{x}{2} = t$ $x/2=t$ , and $\frac{1}{2} d x = d t$ $1/2dx=dt$ .

$= - 2 \int 2 t sin (t) d t = - 4 \int t sin (t) d t$ $=-2int2tsin(t)dt=-4inttsin(t)dt$

Here, use integration by parts, which takes the form

$\int u d v = u v - \int v d u$ $intudv=uv-intvdu$

For $\int t sin (t) d t$ $inttsin(t)dt$ , set $u = t$ $u=t$ and $d v = sin (t) d t$ $dv=sin(t)dt$ , which imply that $d u = d t$ $du=dt$ and $v = - cos (t)$ $v=-cos(t)$ .

Hence,

$\int t sin (t) d t = t (- cos (t)) - \int (- cos (t)) d t$ $inttsin(t)dt=t(-cos(t))-int(-cos(t))dt$

$= - t cos (t) + sin (t) + C$ $=-tcos(t)+sin(t)+C$

Multiply these both by $- 4$ $-4$ to see that:

$- 4 \int t sin (t) d t = 4 t cos (t) - 4 sin (t) + C$ $-4inttsin(t)dt=4tcos(t)-4sin(t)+C$

Now, recall that $t = \frac{x}{2}$ $t=x/2$ :

$\int x sin (\frac{x}{2} - π) d x = 4 (\frac{x}{2}) cos (\frac{x}{2}) - 4 sin (\frac{x}{2}) + C$ $intxsin(x/2-pi)dx=4(x/2)cos(x/2)-4sin(x/2)+C$

$= 2 x cos (\frac{x}{2}) - 4 sin (\frac{x}{2}) + C$ $=2xcos(x/2)-4sin(x/2)+C$

Science

Math

Humanities

... and beyond

What is $\int x sin (\frac{x}{2} - π)$ $int xsin(x/2-pi)$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

What is int xsin(x/2-pi) ∫xsin(x2−π)int xsin(x/2-pi) ?

1 Answer

Explanation:

Related questions

Impact of this question

What is $\int x sin (\frac{x}{2} - π)$ $int xsin(x/2-pi)$ ?