int(cot xdx)/(sqrt(9sin^(2)x-1))

Write Integral: Let's first write down the integral we want to solve. I = ∫(cot x)/(sqrt(9sin^2(x)-1)) dxUse Cot Identity: Notice that cot x = cos x / sin x. Let's rewrite the integral using this identity. I = ∫(cos x / sin x) / (sqrt(9sin^2(x)-1)) dxMake Substitution: To simplify the integral, let's make a substitution. Let u = sin x, which implies du = cos x dx. I = ∫(1/u) / (sqrt(9u^2-1)) duSimplify Integral: Now, we have an integral in terms of u that looks like this: I = ∫(1/u) / (sqrt(9u^2-1)) duConsider Trig Substitution: This integral is not straightforward to solve. We might consider a trigonometric substitution for u, such as u = (1/3)sec(θ), but this would lead to a complex integral that does not simplify easily. Instead, we can recognize that this integral does not have a simple antiderivative in terms of elementary functions. Therefore, we cannot express the indefinite integral in a closed form using elementary functions.

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Find indefinite integrals using the substitution and by parts

Full solution

\int \frac{\cot x}{\sqrt{9\sin^{2}x-1}} \, dx

Write Integral: Let's first write down the integral we want to solve. $\newline$ $I = \int \frac{\cot x}{\sqrt{9\sin^2(x)-1}} \, dx$
Use Cot Identity: Notice that $\cot x = \frac{\cos x}{\sin x}$ . Let's rewrite the integral using this identity. $\newline$ $I = \int \frac{\cos x}{\sin x} / \left(\sqrt{9\sin^2(x)-1}\right) \, dx$
Make Substitution: To simplify the integral, let's make a substitution. Let $u = \sin x$ , which implies $du = \cos x \, dx$ . $\newline$ $I = \int \frac{1}{u} / \left(\sqrt{9u^2-1}\right) \, du$
Simplify Integral: Now, we have an integral in terms of $u$ that looks like this: $\newline$ $I = \int \frac{1/u}{\sqrt{9u^2-1}} \, du$
Consider Trig Substitution: This integral is not straightforward to solve. We might consider a trigonometric substitution for $u$ , such as $u = \frac{1}{3}\sec(\theta)$ , but this would lead to a complex integral that does not simplify easily. Instead, we can recognize that this integral does not have a simple antiderivative in terms of elementary functions. Therefore, we cannot express the indefinite integral in a closed form using elementary functions.