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int[sqrt(tan x)]dx

$\int[\sqrt{\tan x}] d x$

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Find indefinite integrals using the substitution and by parts

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Q.

\int[\sqrt{\tan x}] d x

Simplify Integral Expression: Step $1$ : Simplify the integral expression. $\newline$ We start by recognizing that the integral of the square root of the tangent function is not straightforward. We need to simplify or find a substitution that makes it easier to integrate. $\newline$ Calculation: $\int \sqrt{\tan x}\,dx$
Choose Substitution: Step $2$ : Choose a substitution. $\newline$ Let $u = \tan(x)$ , then $du = \sec^2(x) dx$ . This substitution will transform the integral into a form involving $u$ , which might be easier to integrate. $\newline$ Calculation: $dx = \frac{du}{\sec^2(x)}$
Rewrite Using Substitution: Step $3$ : Rewrite the integral using the substitution. $\newline$ Substituting the values from Step $2$ , we get: $\newline$ $\int[\sqrt{\tan x}]\,dx = \int[\sqrt{u} \cdot (1/\sec^2(x))] \,du$ $\newline$ Since $\sec^2(x) = 1 + \tan^2(x) = 1 + u^2$ , we have: $\newline$ $\int[\sqrt{u} \cdot (1/(1 + u^2))] \,du$
Attempt Integration: Step $4$ : Attempt to integrate the transformed expression. $\newline$ This integral looks complex and might not have a straightforward antiderivative in terms of elementary functions. We attempt integration: $\newline$ $\int \frac{\sqrt{u}}{1 + u^2} \, du$

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