11. Integrals of Inverse, Exponential, & Logarithmic Functions

Using different substitutions

Show that the integral

∫((x² - 1)(x + 1))^(-2/3) dx

can be evaluated with any of the following substitutions.

e. u = tan^(-1) ((x - 1)/2)

What is the value of the integral?

Evaluate the integrals in Exercises 53–76.

73. ∫(from 0 to ln√3) e^x dx/(1+e^(2x))

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫₁² (8 dx / (x² - 2x + 2))

37–56. Integrals Evaluate each integral.

∫ sinh²z dz (Hint: Use an identity.)

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (dx / ((2x + 1)√(4x + 4x²)))

2–9. Integrals Evaluate the following integrals.

∫₀¹ (x² / (9 − x⁶)) dx

Evaluate the definite integral using the appropriate substitutions.

${\int }_{\frac{\pi }{2}}^{\pi }\frac{\sin x}{1+co{s}^{2}x}dx$

Evaluate the integrals in Exercises 53–76.

75. ∫y dy/√(1-y^4)

Evaluate the integrals in Exercises 91–102.

96. ∫dy/((arcsin y)(1-y²))

Evaluate the integrals in Exercises 31–78.

75. ∫(from -2 to -1)2dv/(v²+4v+5)

Evaluate the integrals in Exercises 91–102.

93. ∫(arcsin x)²dx/√(1-x²)

Evaluate the integrals in Exercises 53–76.

59. ∫(from 0 to 1)4ds/√(4-s²)

Evaluate the integrals in Exercises 31–78.

67. ∫(from -2 to 2)3dt/(4+3t²)

Evaluate the integrals in Exercises 77–90.

90. ∫dx/((x-2)√(x²-4x+3))

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.

∫ tan^(-1)(√y) dy