620
Chapter 5 | Integration
5.6 Integrals Involving Exponential and Logarithmic Functions • Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay. • Substitution is often used to evaluate integrals involving exponential functions or logarithms. 5.7 Integrals Resulting in Inverse Trigonometric Functions • Formulas for derivatives of inverse trigonometric functions developed in Derivatives of Exponential and Logarithmic Functions lead directly to integration formulas involving inverse trigonometric functions. • Use the formulas listed in the rule on integration formulas resulting in inverse trigonometric functions to match up
the correct format and make alterations as necessary to solve the problem. • Substitution is often required to put the integrand in the correct form.
CHAPTER 5 REVIEW EXERCISES True or False. Justify your answer with a proof or a counterexample. Assume all functions f and g are continuous over their domains. 439. If f ( x ) >0, f ′( x ) >0 for all x , then the right- hand rule underestimates the integral ∫ a b f ( x ). Useagraph to justify your answer.
1 ⎛
⎝ x 3 −2 x 2 +4 x ⎞
447. ∫
⎠ dx
−1
448. ⌠ ⌡ 0 4
3 t 1+6 t 2
dt
π /2
449. ∫
2sec(2 θ )tan(2 θ ) dθ
b
f ( x ) 2 dx = ∫ a b
b
π /3
440. ∫
f ( x ) dx ∫
f ( x ) dx
a
a
π /4
450. ∫
e cos 2 x sin x cos xdx
f ( x ) ≤ g ( x ) for all x ∈ ⎡
⎤ ⎦ ,
⎣ a , b
441.
If
then
0
b
b
∫ a
f ( x ) ≤ ∫
g ( x ).
Find the antiderivative. 451. ⌠ ⌡ dx ( x +4) 3 452. ∫ x ln ⎛ ⎝ x 2 ⎞ ⎠ dx 453. ⌠ ⌡ 4 x 2 1− x 6 dx 454. ⌠ ⌡ e 2 x 1+ e 4 x dx
a
442. All continuous functions have an antiderivative.
Evaluate the Riemann sums L 4 and R 4 for the following functions over the specified interval. Compare your answer with the exact answer, when possible, or use a calculator to determine the answer. 443. y =3 x 2 −2 x +1 over [−1, 1]
444. y = ln ⎛
⎞ ⎠ over [0, e ]
⎝ x 2 +1
445. y = x 2 sin x over [0, π ]
Find the derivative.
⌠ ⌡ 0 t
446. y = x + 1 x over [1, 4]
455. d dt
sin x 1+ x 2
dx
Evaluate the following integrals.
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