Ch. 8 - Techniques of Integration

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 8 - Techniques of Integration

Problem 8.3.14

Chapter 8, Problem 8.3.14

Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(x) dx

Verified step by step guidance

Recognize that the integral involves the square of the sine function: \(\int_0^{\frac{\pi}{2}} \sin^2(x) \, dx\).

Use the power-reduction identity to rewrite \(\sin^2(x)\) in a more integrable form: \(\sin^2(x) = \frac{1 - \cos(2x)}{2}\).

Substitute this identity into the integral to get \(\int_0^{\frac{\pi}{2}} \frac{1 - \cos(2x)}{2} \, dx\).

Split the integral into two separate integrals: \(\frac{1}{2} \int_0^{\frac{\pi}{2}} 1 \, dx - \frac{1}{2} \int_0^{\frac{\pi}{2}} \cos(2x) \, dx\).

Evaluate each integral separately using basic integration rules, then combine the results to express the value of the original integral.

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Key Concepts

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

Definite Integral

A definite integral calculates the net area under a curve between two specified limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating a definite integral involves finding the antiderivative and then applying the Fundamental Theorem of Calculus.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. For integrals like ∫ sin²(x) dx, using identities such as the power-reduction formula sin²(x) = (1 - cos(2x))/2 simplifies the integral into a more manageable form.

Fundamental Theorem of Calculus

This theorem links differentiation and integration, stating that if F is an antiderivative of f on [a, b], then ∫_a^b f(x) dx = F(b) - F(a). It allows evaluation of definite integrals by finding antiderivatives and computing their difference at the bounds.

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Related Practice

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Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

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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.

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Use any method to evaluate the integrals in Exercises 65–70.

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[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.

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Textbook Question

Evaluate the integrals in Exercises 53–58.

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Textbook Question