Express the following integrals in terms of gamma function(1) โซ ๐๐ฅโ1;๐ฅ410 (2) โซ โcot ๐ ๐๐
Question
Express the following integrals in terms of gamma function
(1)
(2)
Solution
To express the given integrals in terms of the gamma function, we will follow these steps:
(1) โซ ๐๐ฅโ1;๐ฅ410:
Step 1: Rewrite the integral using the gamma function. โซ ๐๐ฅโ1;๐ฅ410 = โซ ๐๐ฅ (๐ฅ^(-1/2)) (๐ฅ^4)^(-1/10)
Step 2: Simplify the expression. โซ ๐๐ฅ (๐ฅ^(-1/2)) (๐ฅ^4)^(-1/10) = โซ ๐๐ฅ (๐ฅ^(-1/2 - 4/10))
Step 3: Apply the properties of the gamma function. โซ ๐๐ฅ (๐ฅ^(-1/2 - 4/10)) = ฮ(1/2 - 4/10 + 1)
Step 4: Simplify the expression further. ฮ(1/2 - 4/10 + 1) = ฮ(1/2 - 2/5 + 1) = ฮ(1/2 + 3/5)
Therefore, the integral โซ ๐๐ฅโ1;๐ฅ410 can be expressed in terms of the gamma function as ฮ(1/2 + 3/5).
(2) โซ โcot ๐ ๐๐:
Step 1: Rewrite the integral using the gamma function. โซ โcot ๐ ๐๐ = โซ (cot ๐)^(1/2) ๐๐
Step 2: Simplify the expression. โซ (cot ๐)^(1/2) ๐๐ = โซ (cos ๐/sin ๐)^(1/2) ๐๐
Step 3: Apply the properties of the gamma function. โซ (cos ๐/sin ๐)^(1/2) ๐๐ = โซ (cos ๐/sin ๐)^(1/2) ๐๐
Step 4: Simplify the expression further. โซ (cos ๐/sin ๐)^(1/2) ๐๐ = โซ (cos ๐/sin ๐)^(1/2) ๐๐
Therefore, the integral โซ โcot ๐ ๐๐ can be expressed in terms of the gamma function as โซ (cos ๐/sin ๐)^(1/2) ๐๐.
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