Question

Evaluate the following integrals: $$ \int \frac{2 x+1}{\sqrt{3+2 x-x^{2}}} d x $$

Short answer

Question: What is the antiderivative of the given integral: $$ \int \frac{2x + 1}{\sqrt{3 + 2x - x^2}} d x $$ Answer: $$ -\frac{4}{3} \int (3+2x-x^2)^{-\frac{1}{2}} (2x-2) dx + \frac{2}{3} \int (3+2x-x^2)^{-\frac{1}{2}} (2x-1) dx - \frac{2}{3}\int (3+2x-x^2)^{-\frac{1}{2}}dx + C $$

Step-by-step solution

Perform Substitution

Perform a substitution to simplify the integral. Let us use the substitution: $$ u = 3 + 2x - x^2 $$ Differentiate with respect to x, we get: $$ d u = (-2 x + 2) d x $$ We now need to express this last equation in terms of \(dx\), so we have: $$ d x = \frac{d u}{-2 x + 2} $$ Now we can rewrite the integral in terms of u: $$ \int \frac{2 x + 1}{\sqrt{3 + 2 x - x^2}} d x = \int \frac{(2 x + 1)(-2 x + 2)}{\sqrt{u}} d u $$

Simplify and evaluate the integral

First, simplify the integral: $$ \int \frac{(2 x + 1)(-2 x + 2)}{\sqrt{u}} d u = -\int \frac{(2x - 1)(2x - 2)}{\sqrt{u}} d u $$ Now it's easier to evaluate the integral. To evaluate the integral, we'll split it into two separate integrals: $$ -\int \frac{(2x - 1)(2x - 2)}{\sqrt{u}} d u = - \int \frac{4x^2 - 4x}{\sqrt{u}} d u + \int \frac{2x - 2}{\sqrt{u}} d u $$ Now we can use the power rule for integration: $$ - \frac{4}{3} \int x^2 u^{-\frac{1}{2}} d u + \frac{2}{3} \int x u^{-\frac{1}{2}} d u - \frac{2}{3}\int u^{-\frac{1}{2}} d u $$ Note that we can use the backward substitution to express the above equation in terms of x: $$ -\frac{4}{3} \int (3+2x-x^2)^{-\frac{1}{2}} (2x-2) dx + \frac{2}{3} \int (3+2x-x^2)^{-\frac{1}{2}} (2x-1) dx - \frac{2}{3}\int (3+2x-x^2)^{-\frac{1}{2}}dx $$

Substitute the original variable back into the antiderivative

The result obtained in step 2 is already in terms of the original variable x. Now we just need to evaluate the individual integrals: $$ -\frac{4}{3} \int (3+2x-x^2)^{-\frac{1}{2}} (2x-2) dx + \frac{2}{3} \int (3+2x-x^2)^{-\frac{1}{2}} (2x-1) dx - \frac{2}{3}\int (3+2x-x^2)^{-\frac{1}{2}}dx = C $$ Where C is the constant of integration. So, the antiderivative of the given integral is: $$ -\frac{4}{3} \int (3+2x-x^2)^{-\frac{1}{2}} (2x-2) dx + \frac{2}{3} \int (3+2x-x^2)^{-\frac{1}{2}} (2x-1) dx - \frac{2}{3}\int (3+2x-x^2)^{-\frac{1}{2}}dx + C $$

Key concepts

Integration by Substitution

One of the most common techniques in solving integrals is called integration by substitution. This method is particularly useful when dealing with complex integrals that can be simplified to a recognizable form. Consider it similar to performing a costume change for the integral, allowing it to be more easily managed.

In the given exercise, substitution is used to transform the original integrand to a function of a new variable, which we chose as \( u = 3 + 2x - x^2 \). We then find the differential of \(u\) in terms of \(x\), which allows us to express \(dx\) as \(\frac{du}{-2x + 2}\). This transformation simplifies the integration process as it reduces the complexity of the function we need to integrate.

Understanding when and how to make a substitution can take some practice, but once mastered, it becomes an invaluable tool in solving integrals efficiently.

Integral Evaluation

Integral evaluation involves finding the antiderivative or the integral of a given function. This process is like a detective work, where you piece together the clues (functions) to find the original 'crime scene' (antiderivative).

The step-by-step solution of our example showcases this process as it simplifies the integral after substitution and splits the complex function into more manageable parts. The decomposition of the integral into smaller fractions eases the task of finding the antiderivatives. Once the integrals are expressed in simpler forms, they can be evaluated separately. Integral evaluation often requires combining multiple techniques and a thorough understanding of fundamental integration rules.

Indefinite Integrals

Indefinite integrals are like the open-ended stories of calculus, representing a family of functions rather than a single value. They include a constant of integration (\(C\)), which is essential as there are infinitely many possible antiderivatives for any given function, each differing by a constant.

In the example provided, the antiderivative is expressed in terms of \(x\), and it encompasses all possible functions whose derivative would give you the original integrand. This general form of the antiderivative is termed as an indefinite integral. Remember, whenever you perform indefinite integration, it's crucial to include the constant of integration \(C\) to acknowledge all potential original functions.

Power Rule for Integration

The power rule for integration is like the reliable friend who is always there to help you out. It is one of the most straightforward integration rules and applies to functions of the form \(x^n\), where \(n eq -1\).

According to the power rule, the integral of \(x^n\) with respect to \(x\) is \(\frac{x^{n+1}}{n+1}\) plus the constant of integration. In our exercise, this rule is utilized after simplifying the integral post-substitution into terms that match the \(u^{-\frac{1}{2}}\) format, which is a variation of \(x^n\). After applying the power rule, the antiderivative becomes much more transparent and straightforward to evaluate.

Applying the Power Rule

To apply the power rule effectively, rewrite the integral in terms of \(u\), where \(u\) is raised to a power, then increment the power by one and divide by the new power. The constant of integration is added last, rounding off the solution.