Question

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

Accepted Answer

Start by making the substitution to simplify the integral. Let $$ u = \sqrt{	heta} $$, which implies $$ 	heta = u^2 . $$Then, differentiate both sides to find $$ d	heta $$: $$ d	heta = 2u \, du . $$Rewrite the integral in terms of $$ u . $$Since $$ \sqrt{	heta} = u $$, the integral becomes $$ \int \frac{\csc^3(u)}{u} d	heta = \int \frac{\csc^3(u)}{u} \cdot 2u \, du = 2 \int \csc^3(u) \, du . $$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 $$ u = \sqrt{	heta}  to $$express the answer in terms of $$ 	heta .$$

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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Key Concepts

Trigonometric Substitution

Reduction Formulas

Integration of Powers of Cosecant