Find the following integrals. (i) ∫x√(x+2)dx x ∈ [0,2]

Question

Find the following integrals.

(i) \(\int\limits_0^2x{\sqrt{x+2}dx}\)

(ii)\(\int\limits_0^{\frac{π}{2}}{\sqrt{sinx}cosxdx}\)

(iii) \(\int\limits_0^{\frac{π}{2}}\frac{1}{4sin^2x + 5cos^2x}\)dx

Answer 1

(i) I =\(\int\limits_0^2x{\sqrt{x+2}dx}\)

Put x + 2 = t² ⇒ dx = 2tdt

When x = 0 ⇒ t = √2, x = 2 ⇒ t = 2

(ii) \(\int\limits_0^{\frac{π}{2}}{\sqrt{sinx}cosxdx}\)

Put sin x = t ⇒ cos xdx = dt

When x = 0 ⇒ t = sin0 = 0,

Put tan x = t ⇒ sec² xdx = dt

When x = 0 ⇒ t = tan 0 = 0,