Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

7. Antiderivatives & Indefinite Integrals

Integrals of Trig Functions

Calculus

7. Antiderivatives & Indefinite Integrals

Integrals of Trig Functions

Previous problemNext problem

2:05 minutes

Problem 8.3.51

Textbook Question

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

Verified step by step guidance

Step 1: Recognize that the integral involves trigonometric functions csc²x and csc⁴x. Break the integral into two separate parts: ∫ csc²x dx + ∫ csc⁴x dx.

Step 2: Recall the standard integral formula for csc²x: ∫ csc²x dx = -cot(x) + C, where C is the constant of integration.

Step 3: For ∫ csc⁴x dx, rewrite csc⁴x as (csc²x)². Use the identity csc²x = 1 + cot²x to express csc⁴x in terms of cot²x.

Step 4: Substitute csc²x = 1 + cot²x into (csc²x)² to get csc⁴x = (1 + cot²x)². Expand this expression to obtain 1 + 2cot²x + cot⁴x.

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.