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Problem 5.4.58a
Bounds on an integral Suppose Ζ is continuous on [a, b] with Ζ''(π) > 0 on the interval. It can be shown that (bβa) Ζ [(a + b) /2] β€ β«βα΅ Ζ(π) dπ β€ (bβa) [ (Ζ(a) + Ζ(b)) /2]
(a) Assuming Ζ is nonnegative on [a, b], draw a figure to illustrate the geometric meaning of these inequalities. Discuss your conclusions. b.
Problem 5.5.20
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« [(βπ + 1)β΄ / 2βπ dπ
Problem 5.5.8
Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating.
β« 8π cos (4πΒ² + 3) dπ, u = 4πΒ² + 3
Problem 5.5.10
Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating.
β« (6π + 1) β(3πΒ² + π) dπ , u = 3πΒ² + π
Problem 5.5.11
Use a substitution of the form u = aπ + b to evaluate the following indefinite integrals.
β«(π + 1)ΒΉΒ² dπ
Problem 5.5.12
Use a substitution of the form u = aπ + b to evaluate the following indefinite integrals
β«(eΒ³Λ£ βΊΒΉ dπ
Problem 5.5.15a
Use Table 5.6 to evaluate the following indefinite integrals.
(a) β« eΒΉβ°Λ£ dπ
Problem 5.5.15b
Use Table 5.6 to evaluate the following indefinite integrals.
(b) β« sec 5π tan 5π dπ
Problem 5.5.15d
Use Table 5.6 to evaluate the following indefinite integrals.
(d) β« cos π/7 dπ
Problem 5.5.15e
Use Table 5.6 to evaluate the following indefinite integrals.
(e) β« dπ/(81 + 9πΒ²) (Hint: Factor a 9 out of the denominator first.)
Problem 5.5.15f
Use Table 5.6 to evaluate the following indefinite integrals.
(f) β« dπ/β36 βπΒ²
Problem 5.5.16c
Use Table 5.6 to evaluate the following definite integrals.
(c) β«βββ^βΆ dπ/(πΒ² β9)
Problem 5.5.16d
Use Table 5.6 to evaluate the following definite integrals.
(d) β«β^Ο/ΒΉβΆ sec Β² 4π dπ
Problem 5.5.17
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« 2π(πΒ² β 1)βΉβΉ dπ
Problem 5.5.18
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« πeΛ£Β² dπ
Problem 5.5.22
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« [ 1/(10πβ3) dπ
Problem 5.5.23
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« πΒ³ (πβ΄ + 16)βΆ dπ
Problem 5.5.26
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« dπ / (β1 β 9πΒ²)
Problem 5.5.29
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« (πβΆ β 3πΒ²)β΄ (πβ΅ β π) dπ
Problem 5.5.32
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« 2 / (πβ4πΒ² β1) dπ , π > Β½
Problem 5.5.35
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« π csc πΒ² cot πΒ² dπ
Problem 5.5.36
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« sec 4w tan 4w dw
Problem 5.5.37
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« secΒ² (10π + 7) dπ
Problem 5.5.40
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« (sinβ΅ π + 3 sinΒ³ πβ sin π) cos π dπ
Problem 5.5.43
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« sin π secβΈ π dπ
Problem 5.5.46
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«βΒΉ 2eΒ²Λ£ dπ
Problem 5.5.49
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«βΒ³ ( 2Λ£ / 2Λ£ + 4 ) dπ
Problem 5.5.54
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«Ο/ββ^Ο/βΈ 8 cscΒ² 4π dπ
Problem 5.5.57
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«Ο/β^Ο/Β² (cos π) / (sinΒ² π) dπ
Problem 5.5.59
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«β/ββ βββ^Β²/β΅ dπ/ xβ(25πΒ²β 1)