9–61. Trigonometric integrals Evaluate the following integrals.
50. ∫ csc¹⁰x cot³x dx

Verified step by step guidance

1. Step 1: Recognize that this is a trigonometric integral involving powers of csc(x) and cot(x). For integrals of this type, it is often helpful to use trigonometric identities to simplify the expression. Recall the identity: $csc²x\; =\; 1\; +\; cot²x$.
2. Step 2: Break down the powers of csc(x) and cot(x) to facilitate substitution. Specifically, split $csc¹⁰x$ into $csc⁸x\; imes\; csc²x$, and use $cot³x$ as is.
3. Step 3: Substitute $u\; =\; cot(x)$, which implies $du\; =\; -csc²x\; dx$. Replace $csc²x\; dx$ with $-du$ in the integral.
4. Step 4: Rewrite the integral in terms of u. Using the substitution, the integral becomes: $\int \; csc⁸x\; imes\; cot³x\; imes\; (-du)$. Replace $csc⁸x$ using the identity $csc²x\; =\; 1\; +\; cot²x$, which leads to powers of u.
5. Step 5: Simplify the integral entirely in terms of u and integrate. After substitution, the integral will involve a polynomial in u. Perform the integration term by term, and then back-substitute $u\; =\; cot(x)$ to express the result in terms of x.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, cosecant, and cotangent, are fundamental in calculus, especially in integration. They describe relationships between angles and sides of triangles and are periodic functions. Understanding their properties, identities, and how they relate to each other is crucial for evaluating integrals involving these functions.

Integration Techniques

Integration techniques, such as substitution and integration by parts, are essential for solving complex integrals. In the case of trigonometric integrals, recognizing patterns and applying appropriate methods can simplify the process. Mastery of these techniques allows for the effective evaluation of integrals that may initially seem daunting.

Pythagorean Identities

Pythagorean identities are equations that relate the squares of trigonometric functions, such as sin²x + cos²x = 1. These identities are useful for transforming and simplifying integrals involving trigonometric functions. Recognizing and applying these identities can help in rewriting integrals in a more manageable form, facilitating easier evaluation.