Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral and check the result by differentiation. $$ \int \frac{x+1}{\left(x^{2}+2 x-3\right)^{2}} d x $$

Step-by-Step Solution

Answer

The indefinite integral of the expression \(\frac{x+1}{\left(x^{2}+2 x-3\right)^{2}}\) is \(-\frac{1}{2(x^{2}+2x-3)}\)

Step 1: Identify a Suitable Substitution

In this case, it is noted that the derivative of the denominator \(x^{2}+2x-3\) is \(2x+2\), which closely resembles the numerator. Hence, the substitution to use is \(u=x^{2}+2x-3\). From this, the derivative \(du = (2x+2)dx\) is calculated.

Step 2: Rewrite the Integral with the Substitution

The integral can now be rewritten using this substitution as \(\int \frac{1}{2} \frac{1}{u^{2}} d u\)

Step 3: Solve the Substituted Integral

The integral simplifies to \(\int \frac{1}{2} u^{-2} du\), which can be resolved to \(-\frac{1}{2u}\).

Step 4: Substitute Back the Value of \(u\)

Replace \(u\) with its original value to get the solution to the indefinite integral. The result is \(-\frac{1}{2(x^{2}+2x-3)}\).

Step 5: Check the Result by Differentiation

Use differentiation to verify the solution. The derivative of \(-\frac{1}{2(x^{2}+2x-3)}\) should produce the original function \(\frac{x+1}{\left(x^{2}+2 x-3\right)^{2}}\) . Remember to use the chain rule for differentiation.

Key Concepts

Indefinite Integral

An indefinite integral represents the antiderivative of a given function. The primary goal is to find a function, say, \( F(x) \), such that its derivative \( F'(x) \) equals the original function. This is written as \( \int f(x) \, dx = F(x) + C \), where \( C \) is the constant of integration. This concept is essential in reversing differentiation and finding functions that describe accumulated quantities over an interval. Here, solving an indefinite integral like \( \int \frac{x+1}{(x^2+2x-3)^2} \, dx \) involves identifying a method that simplifies the integration process, such as substitution.

Substitution Method

The substitution method is a common technique used to simplify difficult integrals by changing variables. This involves replacing a part of the integral with a new variable, \( u \), to make it easier to solve. In our example, observe that the derivative of the denominator, \( x^2 + 2x - 3 \), closely matches the numerator, suggesting substitution is feasible. Let \( u = x^2 + 2x - 3 \). Then, compute its derivative: \( du = (2x+2) \, dx \). Rewriting the integral with these substitutions transforms it into a simpler form: \( \int \frac{1}{2} \cdot \frac{1}{u^2} \, du \). This method is particularly powerful as it can transform unsolvable integrals into elementary ones.

Differentiation

Differentiation is the process of finding a derivative of a function, showing how the function changes at any point. It is the reverse operation of integration. To verify the solution to an integral, differentiation is often used. Once an antiderivative is determined, you can differentiate it to ensure it matches the original function. For example, if the result of an integral is \( -\frac{1}{2(x^2+2x-3)} \), differentiating it should yield the original function \( \frac{x+1}{(x^2 + 2x - 3)^2} \). Calculate this derivative carefully by applying necessary rules, like the chain rule, to confirm accuracy.

Chain Rule

The chain rule is essential for finding the derivative of composite functions, meaning functions of functions. It states that if a function \( y = f(g(x)) \), then its derivative is \( f'(g(x)) \cdot g'(x) \). In checking an indefinite integral by differentiation, the chain rule is often employed. For example, in \( -\frac{1}{2(x^2+2x-3)} \), consider \( x^2 + 2x - 3 \) as \( g(x) \). The derivative requires multiplying by \( g'(x) = 2x + 2 \). By careful application of the chain rule, you ensure differentiation accuracy, which validates your original integral solution. This tool is invaluable for dealing with complex derivatives.