11. Integrals of Inverse, Exponential, & Logarithmic Functions

Evaluate the integral.

${\int }_{}^{}(\frac{2}{\sqrt{1-x^2}}-\frac{1}{3\sqrt{x}})dx$

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

 ∫ 2 / (𝓍√4𝓍² ―1) d𝓍 , 𝓍 > ½ 

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

 ∫₂/₍₅√₃₎^²/⁵ d𝓍/ x√(25𝓍²― 1)

37–56. Integrals Evaluate each integral.

∫₀⁴ sech²√x / √x dx

Find the indefinite integral.

${\int }_{}^{}\frac{1}{x\sqrt{4x^2-16}}dx$

Area of region Find the area of the region bounded by y = sech x, x = 1, and the unit circle (see figure).

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.

∫ (dt / t√(3 + t²)

Evaluate the integrals in Exercises 77–90.

79. ∫(from -1 to 0)6dt/√(3-2t-t²)

Using different substitutions

Show that the integral

∫((x² - 1)(x + 1))^(-2/3) dx

can be evaluated with any of the following substitutions.

c. u = arctan x

What is the value of the integral?

Evaluate the integrals in Exercises 77–90.

81. ∫dy/(y²-2y+5)

37–56. Integrals Evaluate each integral.

∫₀ ˡⁿ ² tanh x dx

2–9. Integrals Evaluate the following integrals.

∫ dx / √(x² − 9),x > 3

Find the indefinite integral.

${\int }_{}^{}\frac{-2}{x^2-12x+45}dx$

Evaluate the integrals in Exercises 1–14.

∫ (2 dx) / (x³ √(x² - 1)), where x > 1

Find the indefinite integral.

${\int }_{}^{}\frac{se{c}^2\left(si{n}^{-1}\theta \right)}{\sqrt{1-{\theta }^2}}d\theta$