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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.1.60
Chapter 8, Problem 8.1.60

7–64. Integration review Evaluate the following integrals.
60. ∫ from 0 to π/4 of 3√(1 + sin 2x) dx

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Step 1: Recognize that the integral ∫ from 0 to π/4 of 3√(1 + sin(2x)) dx involves a composite function. The term √(1 + sin(2x)) suggests that substitution might simplify the integration process.
Step 2: Use the double-angle identity for sine: sin(2x) = 2sin(x)cos(x). This identity may help simplify the integrand or assist in substitution later.
Step 3: Consider substitution to simplify the integral. Let u = 1 + sin(2x). Then, compute the derivative of u with respect to x: du/dx = 2cos(2x). Rewrite dx in terms of du and substitute into the integral.
Step 4: Adjust the limits of integration. When x = 0, calculate the corresponding value of u using u = 1 + sin(2x). Similarly, when x = π/4, calculate the corresponding value of u. Update the limits of integration accordingly.
Step 5: After substitution, the integral should now be in terms of u and simpler to evaluate. Perform the integration with respect to u, and then back-substitute to return to the original variable if necessary.

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Key Concepts

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

Definite Integrals

A definite integral calculates the accumulation of a quantity, represented as the area under a curve, between two specified limits. In this case, the integral from 0 to π/4 indicates that we are interested in the area under the curve of the function 3√(1 + sin 2x) from x = 0 to x = π/4.
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Definition of the Definite Integral

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. In this integral, the expression sin 2x can be simplified using the double angle identity, which states that sin 2x = 2sin x cos x, aiding in the evaluation of the integral.
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Verifying Trig Equations as Identities

Substitution Method

The substitution method is a technique used in integration to simplify the integrand by changing variables. By substituting a new variable for a function of x, the integral can often be transformed into a more manageable form, making it easier to evaluate the definite integral.
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Related Practice
Textbook Question

7–84. Evaluate the following integrals.

45. ∫ from 0 to ln 2 [1 / (1 + eˣ)²] dx

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Textbook Question

79. Tabular integration extended Refer to Exercise 77.

a. The following table shows the method of tabular integration applied to

∫ eˣ cos x dx.

Use the table to express ∫ eˣ cos x dx in terms of the sum of functions and an indefinite integral.

b. Solve the equation in part (a) for ∫ eʳ cos z dz.

c. Evaluate ∫ e⁻ᶻ sin 3z dz by applying the idea from parts (a) and (b).

Textbook Question

23-64. Integration Evaluate the following integrals.

54. ∫ (z + 1)/[z(z² + 4)] dz

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Textbook Question

23-64. Integration Evaluate the following integrals.

41. ∫₋₁¹ x/(x + 3)² dx

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Textbook Question

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

19. ∫ (from 1 to ∞) (3x² + 1)/(x³ + x) dx

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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.

34. ∫ dx / (x(x¹⁰ + 1))