Use partial fractions to find the indefinite integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) ∫9 − x^2/5x^3 + x dx

Answer 1

Assuming the post intends the integrand as [(9-x²)/(5x³+1)], here is the partial fraction decomposition.

First factor the denominator: x (5x² + 1). The binomial is an irreducible quadratic.

(9 - x²)/[x(5x²+1)] = A/x + (Bx + C)/(5x²+1)

9 - x² = A(5x² + 1) + (Bx + C)x

9 - x² = 5Ax² + A + Bx² + Cx

9 - x² = (5A+B)x² + Cx + A

Equate coefficients:

5A + B = -1

A = 9

C = 0

5(9) + B = -1

B=-46

So the original problem can be rewritten as:

∫ [9/x - (46x)/(5x²+1)]dx = ∫ 9dx/x - 46∫(xdx)/(5x²+1)

Antiderivative term by term:

9ln(|x|)

For the 2nd term let u = 5x² + 1

du = 10x dx

-46/10 ∫ du/u

-23/5 ln(5x²+1)

Antiderivative:

9ln(|x|) - (23/5)ln(5x² + 1) + C (absolute value not required in 2nd term because 5x² + 1 always greater than 0).

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