Problem 14 - Calculus: Early Transcendental Functions

Question

Evaluate the indicated integral. $$\int \frac{x+1}{\left(x^{2}+2 x-1\right)^{2}} d x$$

Step-by-Step Solution

Answer

The solution of the integral is: \[ -\frac{1}{2\left( x^{2}+2 x-1\right)} + C \], where \( C \) is the constant of integration.

Step 1: Identify the Suitable Substitution

The first step is to identify the most suitable substitution. A good choice for this problem is \( u= x^{2}+2 x-1 \). Note that the derivative of \( u \) is \( du = (2x + 2)dx = (2(x+1))dx \). This choice of substitution will simplify the integral.

Step 2: Transform the Integral

Now, transform the given integral using the substitution identified in the first step, and express the integral in terms of \( u \). Then we can rewrite the integral as follows: \[\int \frac{(x+1) dx}{(x^{2}+2 x-1)^{2}} = \frac{1}{2}\int\frac{du}{u^2}\]

Step 3: Solve the Integral

Once the integral is transformed, solve it. The antiderivative of \(\frac{1}{u^2}\) is \(-\frac{1}{u}\) so we have: \[\frac{1}{2}\int\frac{1}{u^2} du= -\frac{1}{2u}\]

Step 4: Back-substitute

The final step is to back-substitute \( u \) with the original expression \( x^{2}+2 x-1 \) to obtain the result in terms of \( x \): \[-\frac{1}{2u}= -\frac{1}{2\left( x^{2}+2 x-1\right)}\]

Key Concepts

Substitution Method

When encountering complex integrals, such as those involving powers of binomials, the substitution method can simplify the process. This technique involves changing variables through a substitution to make the integral easier to evaluate.

In the given integral \(\int \frac{x+1}{(x^2 + 2x - 1)^2} \, dx\), we identify \(u = x^2 + 2x - 1\) as our substitution variable. By differentiating \(u\), we find \(du = (2x + 2) \, dx\), which matches up nicely since \(2(x+1) = 2x + 2\). Using this substitution simplifies the integral greatly because it changes the integration variable from \(x\) to \(u\), reflecting a simpler form.

Antiderivative

Finding an antiderivative is central to solving integrals. It is effectively the reverse of taking a derivative. After substituting, our integral changes into a form where finding the antiderivative becomes manageable.

With the integral now as \(\frac{1}{2}\int\frac{du}{u^2}\), the task is to determine the antiderivative of \(\frac{1}{u^2}\). This expression is equivalent to \(u^{-2}\) from which the antiderivative is \(-u^{-1}\) (since the derivative of \(-u^{-1}\) would yield \(\frac{1}{u^2}\)). This step reduces the original complex expression into something straightforward to work with.

Back-substitution

The completion of the integration process reintroduces the original variable by substituting back. This step ensures your solution is in the same terms as your original problem.

After we determine the antiderivative in terms of \(u\), we return to \(x\) by substituting \(u\) back with \(x^2 + 2x - 1\). This restores the variable \(x\) in our expression:

• The antiderivative \(-\frac{1}{2u}\) becomes \(-\frac{1}{2(x^2 + 2x - 1)}\).

Back-substitution is crucial for understanding how the solution directly relates to the original problem.

Polynomial Integral

Polynomial integrals occur frequently in calculus and often require careful manipulation. Expressions involving polynomials, especially under fractions, tend to need simplification through techniques like substitution.

In the given problem, the polynomial is \((x^2 + 2x - 1)^2\). Initially, this suggests a complicated integral due to raised powers. However, reformulating the integral in terms of \(u\) simplifies the process.

• After substitution, the polynomial disappears into the denominator \(u^2\), greatly simplifying the expression.
• This makes the integral straightforward to evaluate, even if the polynomial initially seems daunting.

Understanding polynomial integrals helps in solving them quickly by identifying possible substitutions.