Evaluate (i) ∫ (x+3)(√5-4x-x^2) (ii) ∫ dx/1 + √tanx x ∈ [π/6,π/3]

Question

Evaluate

(i) \(\int{\frac{x+3}{\sqrt{5-42-x^2}}}\)dx

(ii) \(\int\limits_{\frac{π}{6}}^{\frac{π}{3}} \frac{dx}{1+\sqrt{tanx}}\)

Answer 1

(i) put x + 3 = A(-4 - 2x) + B

⇒ -2A = 1 ⇒ A = -\(\frac{1}{2}\)

⇒ -4A + B = 3 ⇒ 2 + B = 3 ⇒ B = 1

Add (1) and (2)