1 / | | x*asin(x) | --------- dx | 3 | / 0
Integral((x*asin(x))/3, (x, 0, 1))
The integral of a constant times a function is the constant times the integral of the function:
Use integration by parts:
Let and let .
Then .
To find :
The integral of is when :
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:
TrigSubstitutionRule(theta=_theta, func=sin(_theta), rewritten=sin(_theta)**2, substep=RewriteRule(rewritten=1/2 - cos(2*_theta)/2, substep=AddRule(substeps=[ConstantRule(constant=1/2, context=1/2, symbol=_theta), ConstantTimesRule(constant=-1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=-cos(2*_theta)/2, symbol=_theta)], context=1/2 - cos(2*_theta)/2, symbol=_theta), context=sin(_theta)**2, symbol=_theta), restriction=(x > -1) & (x < 1), context=x**2/sqrt(1 - x**2), symbol=x)
So, the result is:
So, the result is:
Now simplify:
Add the constant of integration:
The answer is:
/ ________ | / 2 /-1, x < 1) 2 | x*asin(x) \ 2 2 x *asin(x) | --------- dx = C - ------------------------------------------------ + ---------- | 3 6 6 | /
Use the examples entering the upper and lower limits of integration.