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Can u solve integration of √tan x?
Ref: https://edurev.in/question/643092/Can-u-solve-integration-of-√tan-x-

∫√(tan x) dx

Let tan x = t²

⇒ sec² x dx = 2t dt

⇒ dx = [2t / (1 + t⁴)]dt

⇒ Integral ∫ 2t² / (1 + t⁴) dt

⇒ ∫[(t² + 1) + (t² - 1)] / (1 + t⁴) dt

⇒ ∫(t² + 1) / (1 + t⁴) dt + ∫(t² - 1) / (1 + t⁴) dt

⇒ ∫(1 + 1/t² ) / (t² + 1/t² ) dt + ∫(1 - 1/t² ) / (t² + 1/t² ) dt

⇒ ∫(1 + 1/t² )dt / [(t - 1/t)² + 2] + ∫(1 - 1/t²)dt / [(t + 1/t)² -2]

Let t - 1/t = u for the first integral ⇒ (1 + 1/t² )dt = du

and t + 1/t = v for the 2nd integral ⇒ (1 - 1/t² )dt = dv

Integral
= ∫du/(u² + 2) + ∫dv/(v² - 2)

= (1/√2) tan^-1 (u/√2) + (1/2√2) log(v -√2)/(v + √2)l + c

= (1/√2) tan^-1 [(t² - 1)/t√2] + (1/2√2) log (t² + 1 - t√2) / t² + 1 + t√2) + c

= (1/√2) tan^-1 [(tanx - 1)/(√2tan x)] + (1/2√2) log [tanx + 1 - √(2tan x)] / [tan x + 1 + √(2tan x)] + c

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FAQs on Find integral of root tan x . - Class - 12 - Class 12

1. How do I find the integral of √(tan x)?

Ans. To find the integral of √(tan x), you can use the substitution method. Let u = √(tan x), then du = (1/2) sec^2 x dx. By substituting these values into the integral, you can simplify the expression and solve for the integral.

2. Is there a shortcut or formula to find the integral of √(tan x)?

Ans. Unfortunately, there is no specific shortcut or formula to directly find the integral of √(tan x). However, you can use integration techniques such as substitution, trigonometric identities, or integration by parts to simplify the expression and solve for the integral.

3. Can I use a calculator to find the integral of √(tan x)?

Ans. While calculators can be helpful for evaluating definite integrals, they may not be able to find the antiderivative or indefinite integral of √(tan x) directly. You would still need to use integration techniques to solve for the integral symbolically.

4. Are there any specific conditions or restrictions when finding the integral of √(tan x)?

Ans. When finding the integral of √(tan x), it is important to consider the domain of the function. Since tan x is undefined at odd multiples of π/2, you may need to restrict the domain of the integral accordingly. Additionally, be cautious of potential singularities or discontinuities that may affect the integral.

5. Can the integral of √(tan x) be expressed in terms of elementary functions?

Ans. The integral of √(tan x) does not have a simple expression in terms of elementary functions such as polynomials, exponentials, or trigonometric functions. It is considered to be a non-elementary integral. However, it can still be evaluated using integration techniques mentioned earlier.

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