Question

23-64. Integration Evaluate the following integrals.50. ∫ 8(x² + 4)/[x(x² + 8)] dx

Accepted Answer

Start by simplifying the integrand: $$ \frac{8(x^2 + 4)}{x(x^2 + 8)} . $$Notice that the numerator and denominator share some polynomial terms, so try to rewrite the expression to make it easier to integrate. Split the fraction into partial fractions or separate terms if possible. For example, express $$ \frac{8(x^2 + 4)}{x(x^2 + 8)}  as a $$sum of simpler fractions like $$ \frac{A}{x} + \frac{Bx + C}{x^2 + 8} $$, where $$A$$, $$B$$, and $$C$$ are constants to be determined. Multiply both sides of the equation by the denominator $$ x(x^2 + 8)  to $$clear the fractions and set up an equation to solve for $$A$$, $$B$$, and $$C. $$Equate coefficients of corresponding powers of $$x on $$both sides to find these constants. Once you have the constants, rewrite the integral as the sum of integrals of the simpler fractions: $$ \int \frac{A}{x} dx + \int \frac{Bx + C}{x^2 + 8} dx . $$Integrate each term separately using standard integral formulas: $$ \int \frac{1}{x} dx = \ln|x| + C $$, and for $$ \int \frac{Bx + C}{x^2 + a^2} dx $$, use substitution or recognize it as a combination of logarithmic and arctangent integrals.

23-64. Integration Evaluate the following integrals.
50. ∫ 8(x² + 4)/[x(x² + 8)] dx

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Key Concepts

Integration of Rational Functions

Partial Fraction Decomposition

Algebraic Simplification Before Integration