Skip to main content
Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.1.20
Chapter 8, Problem 8.1.20

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.
∫ (dt / t√(3 + t²)

Verified step by step guidance
1
Identify the integral to solve: \(\int \frac{dt}{t \sqrt{3 + t^{2}}}\).
Recognize that the integrand contains \(t\) in the denominator and a square root of a quadratic expression \(3 + t^{2}\). This suggests using a substitution related to \(t^{2}\) or a trigonometric substitution to simplify the square root.
Consider the substitution \(t = \sqrt{3} \tan{\theta}\), which transforms \(3 + t^{2}\) into \(3 + 3 \tan^{2}{\theta} = 3 \sec^{2}{\theta}\). This substitution will simplify the square root expression.
Compute \(dt\) in terms of \(d\theta\): since \(t = \sqrt{3} \tan{\theta}\), then \(dt = \sqrt{3} \sec^{2}{\theta} d\theta\). Substitute \(t\) and \(dt\) back into the integral and simplify the expression.
Rewrite the integral entirely in terms of \(\theta\), simplify the integrand, and then integrate with respect to \(\theta\). After integration, substitute back \(\theta = \arctan{\left( \frac{t}{\sqrt{3}} \right)}\) to express the answer in terms of \(t\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. By letting a new variable represent a function inside the integral, the integral can often be transformed into a more manageable form. This method is especially useful when the integral contains composite functions.
Recommended video:
04:27
Substitution With an Extra Variable

Trigonometric Substitution

Trigonometric substitution replaces algebraic expressions involving square roots with trigonometric functions to simplify integration. For integrals containing expressions like √(a² + x²), substituting x = a tan(θ) can transform the integral into a trigonometric integral that is easier to evaluate.
Recommended video:
6:04
Introduction to Trigonometric Functions

Algebraic Manipulation of Integrals

Algebraic manipulation involves rewriting the integrand using algebraic identities or simplifications before integrating. This can include factoring, rationalizing, or splitting the integral into simpler parts, making the integral more straightforward to solve.
Recommended video:
06:18
Integration by Parts for Definite Integrals
Related Practice
Textbook Question

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ (1 - r²)^(5/2) / r⁸ dr

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 0 to ∞ of (dθ / (1 + e^θ))

1
views
Textbook Question

Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.

∫ z(ln z)² dz

Textbook Question

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.

∫₁^∞ (1 / x^(1/5)) dx

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.

∫ 3 sinh(x/2 + ln 5) dx

1
views
Textbook Question

Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) dx.