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

8. Definite Integrals

Fundamental Theorem of Calculus

Calculus

8. Definite Integrals

Fundamental Theorem of Calculus

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2:10 minutes

Problem 5.3.55

Textbook Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫π/₄^³π/⁴ (cot² 𝓍 + 1) d𝓍

Verified step by step guidance

Step 1: Recognize that the integral ∫π/₄^³π/⁴ (cot² 𝓍 + 1) d𝓍 can be simplified using trigonometric identities. Recall that cot²(𝓍) + 1 = csc²(𝓍). This simplifies the integral to ∫π/₄^³π/⁴ csc²(𝓍) d𝓍.

Step 2: Identify the antiderivative of csc²(𝓍). The antiderivative of csc²(𝓍) is -cot(𝓍). Using this, rewrite the integral as [-cot(𝓍)] evaluated from π/4 to 3π/4.

Step 3: Apply the Fundamental Theorem of Calculus. Substitute the upper limit (𝓍 = 3π/4) and the lower limit (𝓍 = π/4) into the antiderivative -cot(𝓍). This gives -cot(3π/4) - (-cot(π/4)).

Step 4: Simplify the expression. Use the unit circle or trigonometric properties to evaluate cot(3π/4) and cot(π/4). Recall that cot(π/4) = 1 and cot(3π/4) = -1.

Step 5: Combine the results from the previous step to find the value of the definite integral. This involves subtracting the evaluated terms from Step 4.

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Key Concepts

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

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with integration, stating that if a function is continuous on an interval [a, b], then the integral of its derivative over that interval equals the difference in the values of the function at the endpoints. This theorem allows us to evaluate definite integrals by finding an antiderivative of the integrand.

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the limits of integration, which specify the interval, and provides a numerical value that reflects the accumulation of quantities, such as area, over that interval.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. In the context of the given integral, recognizing that cot² x + 1 equals csc² x can simplify the integration process, making it easier to evaluate the integral by transforming it into a more manageable form.