Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.15. ∫ (from 1 to 2) (3x⁵ + 48x³ + 3x² + 16)/(x³ + 16x) dx

Accepted Answer

Step 1: Simplify the integrand by factoring the denominator. The denominator is $$x^3 + 16x$$, which can be factored as $$x(x^2 + 16). $$Rewrite the integrand as $$(3x^5 + 48x^3 + 3x^2 + 16) / (x(x^2 + 16)). $$Step 2: Break the integrand into partial fractions. Express the numerator $$3x^5 + 48x^3 + 3x^2 + 16 as a $$sum of terms that can be divided by $$x(x^2 + 16). $$This involves finding constants $$A, B, C$$ such that $$\frac{A}{x} + \frac{Bx + C}{x^2 + 16}$$ equals the original fraction. Step 3: Solve for the constants $$A, B, C by $$equating coefficients after multiplying through by the denominator $$x(x^2 + 16). $$This step involves algebraic manipulation to match terms on both sides of the equation. Step 4: Integrate each term separately. The term $$\frac{A}{x}$$ integrates to $$A \ln|x|$$, and the term $$\frac{Bx + C}{x^2 + 16}$$ can be split further into simpler integrals using substitution techniques or standard integral formulas for rational functions. Step 5: Evaluate the definite integral by substituting the limits of integration (from $$x = 1 to$$ $$x = 2) $$into the antiderivative obtained in Step 4. Simplify the result to complete the solution.

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
15. ∫ (from 1 to 2) (3x⁵ + 48x³ + 3x² + 16)/(x³ + 16x) dx

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