Integrate (t√t+2 over xcubed

Answer 1

To integrate t√(t+2) / x^3, we can use the substitution method. Let u = t+2, then du = dt.

Therefore, the integral becomes:

∫(t√(t+2) / x^3) dt
= ∫((u-2)√u)/x^3) du
= ∫((u^(3/2) - 2u^(1/2)) / x^3) du
= (1/x^3) ∫(u^(3/2) du - 2/x^3 ∫(u^(1/2) du
= (1/x^3) * (2/5*u^(5/2) - 4/3*u^(3/2)) + C
= (2/(5*x^3)) * (t+2)^(5/2) - (4/(3*x^3)) * (t+2)^(3/2) + C

Therefore, the final integral is (2/(5*x^3)) * (t+2)^(5/2) - (4/(3*x^3)) * (t+2)^(3/2) + C.