Problem 4 - Calculus Early Transcendentals

Question

What is the first step in integrating \(\frac{x^{2}+2 x-3}{x+1} ?\)

Step-by-Step Solution

Answer

The integral of the rational function (x^2 + 2x - 3) / (x + 1) can be found by performing polynomial long division and rewriting the integrand as a proper fraction, then integrating each term separately. The integral is given by: ∫(x - 4 / (x + 1)) dx = (1/2)x^2 - 4ln|x + 1| + C where C is the constant of integration.

Step 1: Perform polynomial long division

Divide the numerator \(x^2 + 2x - 3\) by the denominator \(x + 1\) using polynomial long division. Dividend: \(x^2 + 2x - 3\) Divisor: \(x + 1\) 1. Divide the first term of the dividend \(x^2\) by the first term of the divisor \(x\). The result is \(x\). 2. Multiply the divisor \(x+1\) by \(x\) and write it below the dividend: \(x(x+1) = x^2 + x\). 3. Subtract the obtained result \(x^2 + x\) from the dividend: \((x^2 + 2x - 3) - (x^2 + x) = x - 3\). This is the new dividend. 4. Divide the first term of the new dividend \(x\) by the first term of the divisor \(x\). The result is \(1\). 5. Multiply the divisor \(x + 1\) by \(1\) and write it below the new dividend: \((x+1)(1) = x + 1\). 6. Subtract the obtained result \(x + 1\) from the new dividend: \((x - 3) - (x + 1) = -4\). After performing the polynomial long division, we obtain \(x^2 + 2x - 3 = (x + 1)(x) + (-4)\). So, the integrand can be rewritten as \(x - \frac{4}{x+1}\).

Step 2: Integrate each term separately

Integrate each term of the expression \(x - \frac{4}{x+1}\) with respect to \(x\): \(\int (x - \frac{4}{x + 1}) dx = \int x dx - 4 \int \frac{1}{x + 1} dx\)

Step 3: Evaluate each integral

Evaluate each integral: 1. \(\int x dx = \frac{1}{2}x^2 + C_1\) 2. \(\int \frac{1}{x + 1} dx = \ln|x + 1| + C_2\) So, the integral of the given function is: \(\int (x - \frac{4}{x + 1}) dx = \frac{1}{2}x^2 - 4\ln|x + 1| + C\) where \(C = C_1 + C_2\) is the constant of integration.