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

Indefinite Integrals

Calculus

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

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2:18 minutes

Problem 8.1.68b

Textbook Question

68. Different methods
b. Evaluate ∫(cot x csc² x) dx using the substitution u=cscx.

Verified step by step guidance

Step 1: Begin by identifying the substitution u = csc(x). This substitution simplifies the integral by expressing cot(x) and csc²(x) in terms of u.

Step 2: Compute the derivative of u with respect to x. Since u = csc(x), we know that du/dx = -csc(x)cot(x). Rearrange this to express dx in terms of du: dx = -du / (csc(x)cot(x)).

Step 3: Rewrite the integral ∫(cot(x)csc²(x))dx using the substitution. Replace csc(x) with u and dx with -du / (csc(x)cot(x)). This transforms the integral into ∫(cot(x)u²)(-du / (u cot(x))).

Step 4: Simplify the expression. Notice that cot(x) cancels out, leaving ∫(-u²)du. The integral is now in terms of u, which is easier to evaluate.

Step 5: Integrate -u² with respect to u. The result will be in terms of u, and you can substitute back u = csc(x) to express the final answer in terms of x.

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Key Concepts

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

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fractions. In this case, substitution is particularly useful as it simplifies the integral by changing variables, making it easier to evaluate.

Trigonometric Functions

Trigonometric functions, such as cotangent (cot) and cosecant (csc), are fundamental in calculus, especially in integrals involving angles. Understanding their relationships and identities is crucial for manipulating and simplifying expressions during integration. For instance, csc²(x) is the derivative of -cot(x), which can be leveraged in the integration process.

Substitution Method

The substitution method is a technique in calculus where a new variable is introduced to simplify the integration process. By letting u = csc(x), the integral can be transformed into a more manageable form. This method is particularly effective when the integrand contains a function and its derivative, allowing for easier evaluation of the integral.