Integral of x*a*sin(x/a) dx
The solution
Detail solution
-
Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=ax and let dv(x)=sin(ax).
Then du(x)=a.
To find v(x):
-
Let u=ax.
Then let du=adx and substitute adu:
∫asin(u)du
-
The integral of a constant times a function is the constant times the integral of the function:
∫sin(u)du=a∫sin(u)du
-
The integral of sine is negative cosine:
∫sin(u)du=−cos(u)
So, the result is: −acos(u)
Now substitute u back in:
−acos(ax)
Now evaluate the sub-integral.
-
The integral of a constant times a function is the constant times the integral of the function:
∫(−a2cos(ax))dx=−a2∫cos(ax)dx
-
Let u=ax.
Then let du=adx and substitute adu:
∫acos(u)du
-
The integral of a constant times a function is the constant times the integral of the function:
∫cos(u)du=a∫cos(u)du
-
The integral of cosine is sine:
∫cos(u)du=sin(u)
So, the result is: asin(u)
Now substitute u back in:
asin(ax)
So, the result is: −a3sin(ax)
-
Now simplify:
a2(asin(ax)−xcos(ax))
-
Add the constant of integration:
a2(asin(ax)−xcos(ax))+constant
The answer is:
a2(asin(ax)−xcos(ax))+constant
The answer (Indefinite)
[src]
/
|
| /x\ 3 /x\ 2 /x\
| x*a*sin|-| dx = C + a *sin|-| - x*a *cos|-|
| \a/ \a/ \a/
|
/
∫axsin(ax)dx=C+a3sin(ax)−a2xcos(ax)
/ 2 /1\ /1\\
a*|a *sin|-| - a*cos|-||
\ \a/ \a//
a(a2sin(a1)−acos(a1))
=
/ 2 /1\ /1\\
a*|a *sin|-| - a*cos|-||
\ \a/ \a//
a(a2sin(a1)−acos(a1))
a*(a^2*sin(1/a) - a*cos(1/a))
Use the examples entering the upper and lower limits of integration.