9–61. Trigonometric integrals Evaluate the following integrals.
51. ∫ (csc²x + csc⁴x) dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves trigonometric functions csc²x and csc⁴x. Break the integral into two separate parts: ∫ csc²x dx + ∫ csc⁴x dx.
2. Step 2: Recall the standard integral formula for csc²x: ∫ csc²x dx = -cot(x) + C, where C is the constant of integration.
3. Step 3: For ∫ csc⁴x dx, rewrite csc⁴x as (csc²x)². Use the identity csc²x = 1 + cot²x to express csc⁴x in terms of cot²x.
4. Step 4: Substitute csc²x = 1 + cot²x into (csc²x)² to get csc⁴x = (1 + cot²x)². Expand this expression to obtain 1 + 2cot²x + cot⁴x.
5. Step 5: Break ∫ (1 + 2cot²x + cot⁴x) dx into separate integrals: ∫ 1 dx + ∫ 2cot²x dx + ∫ cot⁴x dx. Solve each integral using appropriate techniques, such as substitution for cot²x and cot⁴x, and combine the results.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and cosecant, are fundamental in calculus, particularly in integration. The cosecant function, csc(x), is defined as the reciprocal of the sine function, csc(x) = 1/sin(x). Understanding these functions is crucial for evaluating integrals involving trigonometric identities.

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and recognizing patterns in integrals. For the integral ∫ (csc²x + csc⁴x) dx, recognizing the forms of csc²x and csc⁴x can help simplify the integration process.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. These identities, such as the Pythagorean identities and reciprocal identities, can simplify integrals. For example, knowing that csc²x = 1 + cot²x can help in breaking down the integral into more manageable parts.