## The following integral requires a preliminary step such as long division or a change of variables before using the method of partial fractio

Question

The following integral requires a preliminary step such as long division or a change of variables before using the method of partial fractions. Evaluate the following integral. x^4 + 7/x^3 + 2x dx Find the partial fraction decomposition of the integrand. x^4 + 7/x^3 + 2x dx

in progress 0
1 year 2021-08-06T15:10:47+00:00 1 Answers 12 views 0

1. Division yields

$$\dfrac{x^4+7}{x^3+2x} = x-\dfrac{2x^2-7}{x^3+2x}$$

Now for partial fractions: you’re looking for constants a, b, and c such that

$$\dfrac{2x^2-7}{x(x^2+2)} = \dfrac ax + \dfrac{bx+c}{x^2+2}$$

$$\implies 2x^2 – 7 = a(x^2+2) + (bx+c)x = (a+b)x^2+cx + 2a$$

which gives a + b = 2, c = 0, and 2a = -7, so that a = -7/2 and b = 11/2. Then

$$\dfrac{2x^2-7}{x(x^2+2)} = -\dfrac7{2x} + \dfrac{11x}{2(x^2+2)}$$

Now, in the integral we get

$$\displaystyle\int\frac{x^4+7}{x^3+2x}\,\mathrm dx = \int\left(x+\frac7{2x} – \frac{11x}{2(x^2+2)}\right)\,\mathrm dx$$

The first two terms are trivial to integrate. For the third, substitute y = x ² + 2 and dy = 2x dx to get

$$\displaystyle \int x\,\mathrm dx + \frac72\int\frac{\mathrm dx}x – \frac{11}4 \int\frac{\mathrm dy}y \\\\ =\displaystyle \frac{x^2}2+\frac72\ln|x|-\frac{11}4\ln|y| + C \\\\ =\displaystyle \boxed{\frac{x^2}2 + \frac72\ln|x| – \frac{11}4 \ln(x^2+2) + C}$$