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

Previous problemNext problem

2:00 minutes

Problem 8.6.85d

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. Using the substitution u = tan(x) in ∫ (tan²x / (tan x - 1)) dx leads to ∫ (u² / (u - 1)) du.

Verified step by step guidance

Step 1: Begin by analyzing the substitution u = tan(x). When using substitution in integration, we replace the original variable (x) with a new variable (u) and also compute the derivative of the substitution. For u = tan(x), the derivative is du/dx = sec²(x), or equivalently, dx = du / sec²(x).

Step 2: Rewrite the original integral ∫ (tan²(x) / (tan(x) - 1)) dx in terms of u. Since tan(x) = u, tan²(x) becomes u², and tan(x) - 1 becomes u - 1. Substitute these expressions into the integral.

Step 3: Replace dx with du / sec²(x) in the integral. The integral now becomes ∫ (u² / (u - 1)) * (1 / sec²(x)) du. At this point, sec²(x) must also be expressed in terms of u to complete the substitution.

Step 4: Recall the trigonometric identity sec²(x) = 1 + tan²(x). Since tan(x) = u, sec²(x) = 1 + u². Substitute this into the integral, replacing sec²(x) with 1 + u².

Step 5: Simplify the integral. After substituting sec²(x) = 1 + u², the integral becomes ∫ (u² / (u - 1)) * (1 / (1 + u²)) du. This does not simplify directly to ∫ (u² / (u - 1)) du, so the statement in the problem is false. The substitution leads to a more complex integral.

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.

Substitution in Integration

Substitution is a technique used in integration to simplify the integrand by changing variables. By letting u = g(x), the integral can be transformed into a function of u, making it easier to evaluate. The differential dx is also converted using the derivative of g(x), which is essential for maintaining the equality of the integral.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. In this context, knowing the identity tan²(x) = sec²(x) - 1 can help simplify the integrand before applying substitution. Understanding these identities is crucial for manipulating expressions involving trigonometric functions effectively.

Limits of Integration

When performing substitution in definite integrals, it is important to change the limits of integration to correspond with the new variable. This ensures that the area under the curve is accurately represented. In indefinite integrals, while limits are not a concern, understanding how they change with substitution is vital for definite integrals.

Watch next

Master Introduction to Indefinite Integrals with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

Evaluate the following integrals.

∫ x/(x² + 6x + 18) dx

Textbook Question

6. Evaluate ∫ cos x √(100 − sin² x) dx using tables after performing the substitution u = sin x.

views

Textbook Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

11. ∫ 3u / (2u + 7) du

views

Textbook Question

15. ∫ x / √(4x + 1) dx

views

Textbook Question

Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.

e. The best approach to evaluating ∫(x³ + 1)/(3x²) dx is to use the change of variables u = x³ + 1.

views

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(g) ∫ ƒ' (g(𝓍))g' (𝓍) d(𝓍) = ƒ(g(𝓍)) + C .

views

Textbook Question

Use Table 5.6 to evaluate the following indefinite integrals.

(a) ∫ e¹⁰ˣ d𝓍

views

Textbook Question

Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals

∫(e³ˣ ⁺¹ d𝓍

views

Show more practices