Problem to solve
Find the indefinite integral \[\int x^2\sqrt{4-3x^2} dx\]
Solution
STEP 1
Take the integral: integral x^2 sqrt(4 - 3 x^2) dx
STEP 2
For the integrand x^2 sqrt(4 - 3 x^2), substitute x = (2 sin(u))/sqrt(3) and dx = (2 cos(u))/sqrt(3) du. Then sqrt(4 - 3 x^2) = sqrt(4 - 4 sin^2(u)) = 2 cos(u) and u = sin^(-1)((sqrt(3) x)/2): = 2/sqrt(3) integral8/3 sin^2(u) cos^2(u) du
STEP 3
Factor out constants: = 16/(3 sqrt(3)) integral sin^2(u) cos^2(u) du
STEP 4
Write cos^2(u) as 1 - sin^2(u): = 16/(3 sqrt(3)) integral sin^2(u) (1 - sin^2(u)) du
STEP 5
Expanding the integrand sin^2(u) (1 - sin^2(u)) gives sin^2(u) - sin^4(u): = 16/(3 sqrt(3)) integral(sin^2(u) - sin^4(u)) du
STEP 6
Integrate the sum term by term and factor out constants: = -16/(3 sqrt(3)) integral sin^4(u) du + 16/(3 sqrt(3)) integral sin^2(u) du
STEP 7
Use the reduction formula, integral sin^m(u) du = -(cos(u) sin^(m - 1)(u))/m + (m - 1)/m integral sin^(-2 + m)(u) du, where m = 4: = (4 sin^3(u) cos(u))/(3 sqrt(3)) + 4/(3 sqrt(3)) integral sin^2(u) du
STEP 8
Write sin^2(u) as 1/2 - 1/2 cos(2 u): = (4 sin^3(u) cos(u))/(3 sqrt(3)) + 4/(3 sqrt(3)) integral(1/2 - 1/2 cos(2 u)) du
STEP 9
Integrate the sum term by term and factor out constants: = (4 sin^3(u) cos(u))/(3 sqrt(3)) - 2/(3 sqrt(3)) integral cos(2 u) du + 2/(3 sqrt(3)) integral1 du
STEP 10
For the integrand cos(2 u), substitute s = 2 u and ds = 2 du: = (4 sin^3(u) cos(u))/(3 sqrt(3)) - 1/(3 sqrt(3)) integral cos(s) ds + 2/(3 sqrt(3)) integral1 du
STEP 11
The integral of cos(s) is sin(s): = -sin(s)/(3 sqrt(3)) + (4 sin^3(u) cos(u))/(3 sqrt(3)) + 2/(3 sqrt(3)) integral1 du
STEP 12
The integral of 1 is u: = -sin(s)/(3 sqrt(3)) + (2 u)/(3 sqrt(3)) + (4 sin^3(u) cos(u))/(3 sqrt(3)) + constant
STEP 13
Substitute back for s = 2 u: = (2 u)/(3 sqrt(3)) - sin(2 u)/(3 sqrt(3)) + (4 sin^3(u) cos(u))/(3 sqrt(3)) + constant
STEP 14
Apply the double angle formula sin(2 u) = 2 sin(u) cos(u): = (2 u)/(3 sqrt(3)) + (4 sin^3(u) cos(u))/(3 sqrt(3)) - (2 sin(u) cos(u))/(3 sqrt(3)) + constant
STEP 15
Express cos(u) in terms of sin(u) using cos^2(u) = 1 - sin^2(u): = (2 u)/(3 sqrt(3)) - (2 sqrt(1 - sin^2(u)) sin(u))/(3 sqrt(3)) + (4 sin^3(u) cos(u))/(3 sqrt(3)) + constant
STEP 16
Substitute back for u = sin^(-1)((sqrt(3) x)/2): = -1/6 sqrt(4 - 3 x^2) x + 1/4 sqrt(4 - 3 x^2) x^3 + (2 sin^(-1)((sqrt(3) x)/2))/(3 sqrt(3)) + constant
STEP 17
Which is equal to: Answer: | | = 1/36 (3 x sqrt(4 - 3 x^2) (3 x^2 - 2) + 8 sqrt(3) sin^(-1)((sqrt(3) x)/2)) + constant
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Edited by tododemate® - 2021