Question

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.∫₀^∞ dx / [(x + 1)(x² + 1)]

Accepted Answer

Start by expressing the integrand $$ \frac{1}{(x+1)(x^2+1)}  as a $$sum of partial fractions. Set up the equation: $$ \frac{1}{(x+1)(x^2+1)} = \frac{A}{x+1} + \frac{Bx + C}{x^2 + 1} $$ where $$A$$, $$B$$, and $$C$$ are constants to be determined. Multiply both sides of the equation by $$ (x+1)(x^2+1)  to $$clear the denominators, resulting in: $$ 1 = A(x^2 + 1) + (Bx + C)(x + 1) . $$Expand the right-hand side and collect like terms in powers of $$x. $$Equate the coefficients of corresponding powers of $$x on $$both sides to form a system of equations for $$A$$, $$B$$, and $$C. $$Solve this system to find the values of these constants. Rewrite the integral as the sum of integrals of the partial fractions: $$ \int_0^\infty \frac{dx}{(x+1)(x^2+1)} = \int_0^\infty \frac{A}{x+1} dx + \int_0^\infty \frac{Bx + C}{x^2 + 1} dx . $$Evaluate each integral separately. For $$ \int \frac{1}{x+1} dx $$, use the natural logarithm function. For $$ \int \frac{x}{x^2 + 1} dx $$, use substitution $$ u = x^2 + 1 . $$For $$ \int \frac{1}{x^2 + 1} dx $$, use the arctangent function. Then, apply the limits from 0 to $$ \infty $$ carefully to find the value of the original integral.

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ dx / [(x + 1)(x² + 1)]

Verified video answer for a similar problem:

Key Concepts

Improper Integrals

Partial Fraction Decomposition

Integration of Rational Functions