11. Integrals of Inverse, Exponential, & Logarithmic Functions

Evaluate the integral.

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

Evaluate the integral.

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

Find the indefinite integral.

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

Evaluate the definite integral in terms of an inverse trig function.

$\int_0^{\frac{\sqrt3}{3}}\frac{dx}{\sqrt{4-9x^2}}$

Evaluate the definite integral using the appropriate substitutions.

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

Find the indefinite integral.

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

Find the indefinite integral.

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

Find the indefinite integral.

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

76. Apparent discrepancy

Three different computer algebra systems give the following results:

∫ (dx / (x√(x⁴ − 1))) = ½ cos⁻¹(√(x⁻⁴)) = ½ cos⁻¹(x⁻²) = ½ tan⁻¹(√(x⁴ − 1)).

Explain how all three can be correct.

Use Table 5.6 to evaluate the following definite integrals.                                                                                                                    

 (c) ∫₃√₂^⁶ d𝓍/(𝓍² ―9)

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 

Evaluating integrals Evaluate the following integrals.

∫√₂/₅^²/⁵ d𝓍/𝓍√(25𝓍² ―1)

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)