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Evaluate the integral. 8 dx √2 f (x2-1)3/2 Step 1 Recall the Inverse Substitution Rule, where f and g are differentiable functions and g is one-to-one. Â √x) dx = fg dx = [f(g(t)g'(t) f(g(t))g'(t) dt We are given the following. 8 dx (x² - 1)3/2 We note that (x² - 1) 3/2 = (√√x² - 1)³. 3 Expression x2a2 x = a sec(0), 3 If (√√x²-1)³ = (√√x²-a)³. 3 then a = Therefore, the following entry from the Table of Trigonometric Substitutions is appropriate. Substitution Identity 0 ≤ 0 <

Evaluate the integral. 8 dx √2 f (x2-1)3/2 Step 1 Recall the Inverse Substitution Rule, where f and g are differentiable functions and g is one-to-one. Â √x) dx = fg dx = [f(g(t)g'(t) f(g(t))g'(t) dt We are given the following. 8 dx (x² - 1)3/2 We note that (x² - 1) 3/2 = (√√x² - 1)³. 3 Expression x2a2 x = a sec(0), 3 If (√√x²-1)³ = (√√x²-a)³. 3 then a = Therefore, the following entry from the Table of Trigonometric Substitutions is appropriate. Substitution Identity 0 ≤ 0 <

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Evaluate the integral. 8 dx √2 f (x2-1)3/2 Step 1 Recall the Inverse Substitution Rule, where f and g are differentiable functions and g is one-to-one. Â √x) dx = fg dx = [f(g(t)g'(t) f(g(t))g'(t) dt We are given the following. 8 dx (x² - 1)3/2 We note that (x² - 1) 3/2 = (√√x² - 1)³. 3 Expression x2a2 x = a sec(0), 3 If (√√x²-1)³ = (√√x²-a)³. 3 then a = Therefore, the following entry from the Table of Trigonometric Substitutions is appropriate. Substitution Identity 0 ≤ 0 < <or π ≤ 0<sec²(0) - 1 = tan² (0) 2 2 Therefore, we can let x = sec(0), so dx = We also must make a substitution for the limits of integration in the definite integral. Since x x = 8,0 = sec ]). do. πT = sec(0), we note that when x = 2,0 = Further, when
Transcribed Image Text:Evaluate the integral. 8 dx √2 f (x2-1)3/2 Step 1 Recall the Inverse Substitution Rule, where f and g are differentiable functions and g is one-to-one. Â √x) dx = fg dx = [f(g(t)g'(t) f(g(t))g'(t) dt We are given the following. 8 dx (x² - 1)3/2 We note that (x² - 1) 3/2 = (√√x² - 1)³. 3 Expression x2a2 x = a sec(0), 3 If (√√x²-1)³ = (√√x²-a)³. 3 then a = Therefore, the following entry from the Table of Trigonometric Substitutions is appropriate. Substitution Identity 0 ≤ 0 < <or π ≤ 0<sec²(0) - 1 = tan² (0) 2 2 Therefore, we can let x = sec(0), so dx = We also must make a substitution for the limits of integration in the definite integral. Since x x = 8,0 = sec ]). do. πT = sec(0), we note that when x = 2,0 = Further, when
O O Evaluate the integral using the indicated trigonometric substitution. (Use C for the constant of integration.) +3 dx, x = 5 tan(0) 25 + x2 Sketch and label the associated right triangle. √√x²+25 0 5 ⑪O Ꮎ +3 5 √x² x² +25 ⑪O Ꮎ √√x² + 2 +25 X e 5 5 X x²+25
Transcribed Image Text:O O Evaluate the integral using the indicated trigonometric substitution. (Use C for the constant of integration.) +3 dx, x = 5 tan(0) 25 + x2 Sketch and label the associated right triangle. √√x²+25 0 5 ⑪O Ꮎ +3 5 √x² x² +25 ⑪O Ꮎ √√x² + 2 +25 X e 5 5 X x²+25
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