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. 

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

Evaluate the integral. 

${\int }_{}^{}\frac{4x^2+2x+3}{x^2+x}dx$

Find the area under the curve of $y=\frac{10}{(x+2)(x+3)}$ between $x=0$ and $x=4$.

Express the integrand as a sum of partial fractions and evaluate the integral.

${\int }_{}^{}\frac{2}{x^3+x^2-x-1}dx$

Use the method of partial fractions to evaluate the integral.

${\int }_{}^{}\frac{5x^3-x^2+7x-2}{(x^2+1)(x^2+2)}dx$

Evaluate the integral.

${\int }_{}^{}\frac{t^4+t^2-2}{t^3+t}dt$

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

21. ∫ cos x / (sin² x + 2 sin x) dx

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

34. ∫ dx / (x(x¹⁰ + 1))