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Chapter 7: Problem 4

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

Short Answer

Expert verified
Answer: \(\frac{x^2}{2} - x + C\)

Step by step solution

01

Perform Partial Fraction Decomposition

To perform partial fraction decomposition, we must first factor the denominator: \(x+1\). Since it's linear, it cannot be factored further. Now we express the given rational function as the sum of two partial fractions: $$\frac{x^{2}+2 x-3}{x+1} = \frac{A}{x+1} + B$$ Now multiply both sides by the common denominator \((x+1)\): $$x^{2}+2 x-3 = A(x+1) + B(x+1)(x-1)$$
02

Solve for A and B

To solve for A and B, we can plug in values of x that make one of the terms on the right side equal to 0. Let \(x = -1\), we get: $$(-1)^2 + 2(-1) - 3 = A(-1+1)$$ $$2-2-3 = 0$$ So, \(A = 0\). $$\Rightarrow x^2+2x-3 = B(x+1)(x-1)$$ Now we can compare the coefficients on both sides of this equation: Equating the x虏 coefficients, we get: $$1 = B(1)$$ $$B=1$$
03

Rewrite the function using Partial Fraction Decomposition

We rewrite the original rational function using the coefficients we found: $$\frac{x^{2}+2 x-3}{x+1} = \frac{1}{x+1}(x-1)$$ $$\frac{x^{2}+2 x-3}{x+1} = x-1$$
04

Perform the integration

Now we will integrate the function: $$\int \frac{x^{2}+2 x-3}{x+1} dx = \int (x-1) dx$$ Applying the power rule for integration, we get: $$\int (x-1) dx = \frac{x^2}{2} - x + C$$ The first step in integrating \(\frac{x^{2}+2 x-3}{x+1}\) is to perform partial fraction decomposition and integrate term by term. So, the answer is: \(\frac{x^2}{2} - x + C\).

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