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Express the following integrals in terms of gamma function(1) โˆซ ๐‘‘๐‘ฅโˆš1;๐‘ฅ410 (2) โˆซ โˆšcot ๐œƒ ๐‘‘๐œƒ

Question

Express the following integrals in terms of gamma function

(1) โˆซdx1+x4 \int \frac{dx}{\sqrt{1+x^4}}

(2) โˆซcotโก(ฮธ)โ€‰dฮธ \int \sqrt{\cot(\theta)} \, d\theta

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Solution

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

(1) โˆซ ๐‘‘๐‘ฅโˆš1;๐‘ฅ410:

Step 1: Rewrite the integral using the gamma function. โˆซ ๐‘‘๐‘ฅโˆš1;๐‘ฅ410 = โˆซ ๐‘‘๐‘ฅ (๐‘ฅ^(-1/2)) (๐‘ฅ^4)^(-1/10)

Step 2: Simplify the expression. โˆซ ๐‘‘๐‘ฅ (๐‘ฅ^(-1/2)) (๐‘ฅ^4)^(-1/10) = โˆซ ๐‘‘๐‘ฅ (๐‘ฅ^(-1/2 - 4/10))

Step 3: Apply the properties of the gamma function. โˆซ ๐‘‘๐‘ฅ (๐‘ฅ^(-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 โˆซ ๐‘‘๐‘ฅโˆš1;๐‘ฅ410 can be expressed in terms of the gamma function as ฮ“(1/2 + 3/5).

(2) โˆซ โˆšcot ๐œƒ ๐‘‘๐œƒ:

Step 1: Rewrite the integral using the gamma function. โˆซ โˆšcot ๐œƒ ๐‘‘๐œƒ = โˆซ (cot ๐œƒ)^(1/2) ๐‘‘๐œƒ

Step 2: Simplify the expression. โˆซ (cot ๐œƒ)^(1/2) ๐‘‘๐œƒ = โˆซ (cos ๐œƒ/sin ๐œƒ)^(1/2) ๐‘‘๐œƒ

Step 3: Apply the properties of the gamma function. โˆซ (cos ๐œƒ/sin ๐œƒ)^(1/2) ๐‘‘๐œƒ = โˆซ (cos ๐œƒ/sin ๐œƒ)^(1/2) ๐‘‘๐œƒ

Step 4: Simplify the expression further. โˆซ (cos ๐œƒ/sin ๐œƒ)^(1/2) ๐‘‘๐œƒ = โˆซ (cos ๐œƒ/sin ๐œƒ)^(1/2) ๐‘‘๐œƒ

Therefore, the integral โˆซ โˆšcot ๐œƒ ๐‘‘๐œƒ can be expressed in terms of the gamma function as โˆซ (cos ๐œƒ/sin ๐œƒ)^(1/2) ๐‘‘๐œƒ.

This problem has been solved

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