Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ 1 / (x√x + 9) dx
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.3.50
Evaluate the integrals in Exercises 33–52.
∫ 8 cot⁴(t) dt
Verified step by step guidance1
Recall that \( \cot(t) = \frac{\cos(t)}{\sin(t)} \) and that powers of cotangent can be expressed in terms of cosecant using the identity \( \cot^2(t) = \csc^2(t) - 1 \). This will help simplify the integral.
Rewrite \( \cot^4(t) \) as \( (\cot^2(t))^2 \), then use the identity to express it in terms of \( \csc^2(t) \):
\[
8 \cot^4(t) = 8 (\cot^2(t))^2 = 8 (\csc^2(t) - 1)^2
\]
Expand the square to get:
\[
8 (\csc^4(t) - 2 \csc^2(t) + 1)
\]
which breaks the integral into three simpler integrals:
Set up the integral as the sum of three integrals:
\[
\int 8 \cot^4(t) \, dt = \int 8 \csc^4(t) \, dt - \int 16 \csc^2(t) \, dt + \int 8 \, dt
\]
Evaluate each integral separately:
- For \( \int \csc^2(t) \, dt \), recall that the derivative of \( -\cot(t) \) is \( \csc^2(t) \).
- For \( \int \csc^4(t) \, dt \), use reduction formulas or rewrite \( \csc^4(t) = (\csc^2(t))^2 \) and express in terms of \( \cot(t) \) and \( \csc(t) \) to integrate.
- The integral of a constant is straightforward.
Combine all results to express the final integral.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. For integrating powers of cotangent, identities like cot²(t) = csc²(t) - 1 help simplify the integrand into more manageable terms.
Recommended video:
7:17
Verifying Trig Equations as Identities
Integration of Powers of Trigonometric Functions
Integrating powers of trigonometric functions often requires rewriting the integrand using identities or reduction formulas. For cotangent raised to a power, expressing cot⁴(t) in terms of cot²(t) and then using substitution or known integrals facilitates the evaluation.
Recommended video:
6:04Introduction to Trigonometric Functions
Substitution Method in Integration
The substitution method involves changing variables to simplify an integral. When integrating functions like cotangent powers, substituting u = cot(t) or using related expressions can transform the integral into a polynomial form, making it easier to solve.
Recommended video:
07:33Euler's Method
Related Practice
Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^π 8cos⁴(2πx) dx
1
views
Textbook Question
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from -1 to 1 of (-x ln|x| dx)
1
views
Textbook Question
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ (from 0 to √3/2) dy / (1 - y²)^(5/2)
1
views
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (2 ln(z³)) / (16z) dz
1
views
Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ 1 / (x⁶(x⁵ + 4)) dx