Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ csc³(√θ) / √θ dθ

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Start by making the substitution to simplify the integral. Let \( u = \sqrt{\theta} \), which implies \( \theta = u^2 \). Then, differentiate both sides to find \( d\theta \): \( d\theta = 2u \, du \).

Rewrite the integral in terms of \( u \). Since \( \sqrt{\theta} = u \), the integral becomes \( \int \frac{\csc^3(u)}{u} d\theta = \int \frac{\csc^3(u)}{u} \cdot 2u \, du = 2 \int \csc^3(u) \, du \). Notice how the \( u \) terms cancel out.

Focus now on evaluating \( \int \csc^3(u) \, du \). This is a standard integral that can be solved using a reduction formula or by expressing \( \csc^3(u) \) as \( \csc(u) \cdot \csc^2(u) \) and using integration by parts.

Recall the reduction formula for \( \int \csc^n(u) \, du \) when \( n \) is an odd integer: \[ \int \csc^n(u) \, du = -\frac{\csc^{n-2}(u) \cot(u)}{n-1} + \frac{n-2}{n-1} \int \csc^{n-2}(u) \, du \]. Apply this formula with \( n=3 \) to reduce the integral to a simpler form.

After applying the reduction formula, integrate the resulting simpler integral(s) and then substitute back \( u = \sqrt{\theta} \) to express the answer in terms of \( \theta \).

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Trigonometric Substitution

Trigonometric substitution involves replacing a variable or expression with a trigonometric function to simplify integrals, especially those involving roots or powers of trigonometric functions. It helps transform complicated integrals into more manageable forms by leveraging trigonometric identities.

Reduction Formulas

Reduction formulas are recursive relationships that express an integral with a higher power in terms of an integral with a lower power. They simplify the evaluation of integrals involving powers of trigonometric functions by breaking them down step-by-step.

Integration of Powers of Cosecant

Integrating powers of cosecant functions often requires using identities and reduction formulas due to their complexity. Understanding how to manipulate expressions like csc³(x) and apply appropriate substitutions is essential for solving such integrals.

Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.∫ csc³(√θ) / √θ dθ