12. Techniques of Integration

7–84. Evaluate the following integrals.

45. ∫ from 0 to ln 2 [1 / (1 + eˣ)²] dx

Express the rational function as a sum or difference of two simpler fractions. Use a system of equations.

$\frac{1}{(2x+1)(2x-1)}$

Express the rational function as a sum or difference of two simpler fractions. Use a system of equations.

$\frac{4x^2-21x-40}{x^3-x^2-20x}$

Give the partial fraction decomposition for the following expression using strategic substitutions for $x$.

$\frac{4x^2-8x-8}{x(x-2)(x+2)}$

Express the rational function as a sum or difference or simpler rational expressions.

$\frac{4}{x^3-3x^2-x+3}$

Express the rational function as a sum or difference or simpler rational expressions.

$\frac{13x+2}{2x^2-x-1}$

Evaluate the integral. 

${\int }_{}^{}\frac{-6x^2+3x+5}{x^3-x}dx$

Evaluate the integral. 

${\int }_0^3\frac{3x+10}{x^2+9x+20}dx$