Question

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.∫ (3t² + t + 4) / (t³ + t) dt from 1 to √3

Accepted Answer

First, factor the denominator of the integrand. The denominator is $$t^{3} + t$, which can be factored as t(t$$^{2}$$ + 1). $$Next, express the integrand $$\frac{3t$$^{2}$$ + t + 4}{t(t$$^{2}$$ + 1)} as a $$sum of partial fractions. Since the denominator factors into a linear term $$t$$ and an irreducible quadratic $$t^{2} + 1$, set up the decomposition as: $$\frac{$$3t^{2} + t + 4$$}{$$t(t^{2} + 1$$)} = \frac{A}{t} + \frac{Bt + C}{$$t^{2} + 1$$}$$, where A$, B$, and C$ are constants to be determined. Multiply both sides of the equation by the denominator t(t$$^{2}$$ + 1) to $$clear the fractions, resulting in: $$3t^{2} + t + 4$$ = $$A(t^{2} + 1) + (Bt + C)t$. Expand the right-hand side and group like terms: 3t$$^{2}$$ + t + 4 = A t$$^{2}$$ + A + B t$$^{2}$$ + C t = (A + B) t$$^{2}$$ + C t + A. $$Then, equate the coefficients of corresponding powers of $$t on $$both sides to form a system of equations: for $$t^{2}$$, $$3 = A + B$$; for $$t$$, $$1 = C$$; for the constant term, $$4 = A. $$Solve the system of equations to find the values of $$A$$, $$B$$, and $$C. $$Once these constants are found, rewrite the integrand as the sum of partial fractions and then integrate each term separately over the interval from $$1 to$$ $$\sqrt{3}.$$

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (3t² + t + 4) / (t³ + t) dt from 1 to √3

Verified video answer for a similar problem:

Key Concepts

Partial Fraction Decomposition

Integration of Rational Functions

Definite Integrals and Limits of Integration