2 / | | __________ | / 2 | \/ 8 - 2*x dx | / -2
Integral(sqrt(8 - 2*x^2), (x, -2, 2))
Rewrite the integrand:
The integral of a constant times a function is the constant times the integral of the function:
TrigSubstitutionRule(theta=_theta, func=2*sin(_theta), rewritten=4*cos(_theta)**2, substep=ConstantTimesRule(constant=4, other=cos(_theta)**2, substep=RewriteRule(rewritten=cos(2*_theta)/2 + 1/2, substep=AddRule(substeps=[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), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(2*_theta)/2 + 1/2, symbol=_theta), context=cos(_theta)**2, symbol=_theta), context=4*cos(_theta)**2, symbol=_theta), restriction=(x > -2) & (x < 2), context=sqrt(4 - x**2), symbol=x)
So, the result is:
Now simplify:
Add the constant of integration:
The answer is:
/ | | __________ // ________ \ | / 2 ___ || / 2 | | \/ 8 - 2*x dx = C + \/ 2 *|< /x\ x*\/ 4 - x | | ||2*asin|-| + ------------- for And(x > -2, x < 2)| / \\ \2/ 2 /
Use the examples entering the upper and lower limits of integration.