Express the following integrals in terms of gamma function(1) ∫ 𝑑𝑥√1;𝑥410 (2) ∫ √cot 𝜃 𝑑𝜃

To express the given integrals in terms of the gamma function, we will follow these steps:

(1) ∫ 𝑑𝑥√1;𝑥410:

Step 1: Rewrite the integral using the gamma function. ∫ 𝑑𝑥√1;𝑥410 = ∫ 𝑑𝑥 (𝑥^(-1/2)) (𝑥^4)^(-1/10)

Step 2: Simplify the expression. ∫ 𝑑𝑥 (𝑥^(-1/2)) (𝑥^4)^(-1/10) = ∫ 𝑑𝑥 (𝑥^(-1/2 - 4/10))

Step 3: Apply the properties of the gamma function. ∫ 𝑑𝑥 (𝑥^(-1/2 - 4/10)) = Γ(1/2 - 4/10 + 1)

Step 4: Simplify the expression further. Γ(1/2 - 4/10 + 1) = Γ(1/2 - 2/5 + 1) = Γ(1/2 + 3/5)

Therefore, the integral ∫ 𝑑𝑥√1;𝑥410 can be expressed in terms of the gamma function as Γ(1/2 + 3/5).

(2) ∫ √cot 𝜃 𝑑𝜃:

Step 1: Rewrite the integral using the gamma function. ∫ √cot 𝜃 𝑑𝜃 = ∫ (cot 𝜃)^(1/2) 𝑑𝜃

Step 2: Simplify the expression. ∫ (cot 𝜃)^(1/2) 𝑑𝜃 = ∫ (cos 𝜃/sin 𝜃)^(1/2) 𝑑𝜃

Step 3: Apply the properties of the gamma function. ∫ (cos 𝜃/sin 𝜃)^(1/2) 𝑑𝜃 = ∫ (cos 𝜃/sin 𝜃)^(1/2) 𝑑𝜃

Step 4: Simplify the expression further. ∫ (cos 𝜃/sin 𝜃)^(1/2) 𝑑𝜃 = ∫ (cos 𝜃/sin 𝜃)^(1/2) 𝑑𝜃

Therefore, the integral ∫ √cot 𝜃 𝑑𝜃 can be expressed in terms of the gamma function as ∫ (cos 𝜃/sin 𝜃)^(1/2) 𝑑𝜃.