How do you integrate x^3/sqrt(144-x^2)?

1 Answer
Aug 23, 2015

Use substitution with u = 144-x^2.

Explanation:

int x^3/sqrt(144-x^2) dx

Let u = 144-x^2, so that du = -2xdx and x^2 = 144-u

int x^3/sqrt(144-x^2) dx = int x^2/sqrt(144-x^2)" " xdx

= -1/2 int x^2/sqrt(144-x^2)" "(-2 x)dx

= -1/2 int (144- u)/sqrtu" "du

= -1/2 int (144/sqrtu - sqrtu)" "du

= -1/2 int (144u^(-1/2) -u^(1/2))" "du

= -1/2[288u^(1/2)-2/3u^(3/2)] +C

= -144(144-x^2)^(1/2)+ 1/3(144-x^2)^(3/2) +C

= -144sqrt(144-x^2)+ 1/3(sqrt(144-x^2))^3 +C