Question

Evaluate the following integrals: $$ \int \sqrt{2 a x+x^{2}} d x $$

Short answer

Question: Evaluate the integral: \(\int \sqrt{2 a x + x^{2}}\, d x.\) Answer: The integral evaluates to \(\frac{1}{2}(x + a)\sqrt{(x + a)^{2} - a^{2}} + \frac{a^2}{2}\arcsin\left(\frac{x + a}{a}\right) + C.\)

Step-by-step solution

Identify the substitution

Notice that the expression under the square root can be rewritten as the square of a binomial: \(2 a x + x^{2} = (x + a)^{2} - a^{2}.\) We'll use the following substitution: $$ u = x + a. $$ Now, we need to find the differential \(d x\) in terms of \(d u.\) From the substitution, we get \(x = u - a.\) Thus, we have: $$ d x = d u. $$

Rewrite the integral using the substitution

Replace \(x\) and \(d x\) in the original integral with expressions in terms of \(u\): $$ \int \sqrt{2 a x + x^{2}}\, d x = \int \sqrt{(u - a) + a^2 - a^2}\, d u = \int \sqrt{u^2 - a^2}\, d u $$

Solve the integral

We can recognize the expression under the square root as the case of the integral of \(\sqrt{a^2 - x^2}.\) Thus, we can use the formula: $$ \int \sqrt{a^2 - x^2}\, d x = \frac{1}{2}x\sqrt{a^2 - x^2} + \frac{a^2}{2}\arcsin\left(\frac{x}{a}\right) + C $$ Hence, we have: $$ \int \sqrt{u^2 - a^2}\, d u = \frac{1}{2}u\sqrt{u^2 - a^2} + \frac{a^2}{2}\arcsin\left(\frac{u}{a}\right) + C. $$

Replace back the variable x

Plug back the substitution \(u = x + a\) into our result: $$ \int \sqrt{2 a x + x^{2}}\, d x = \frac{1}{2}(x + a)\sqrt{(x + a)^{2} - a^{2}} + \frac{a^2}{2}\arcsin\left(\frac{x + a}{a}\right) + C. $$ This is the final solution for the given integral.

Key concepts

Integration by Substitution

The technique of integration by substitution is akin to the reverse process of the chain rule in differentiation. It's a powerful method to simplify integrals by replacing complex expressions with a single variable, usually denoted as 'u'.

To apply this method, one must first identify a portion of the integral that can be considered as a single entity, then replace it with 'u'. This part of the integral is often the inner function of a composition of functions. After the substitution, we replace the differential 'dx' with 'du', based on the derivative of 'u' with respect to 'x'.

• Choose a substitution to simplify the integral
• Replace all occurrences of 'x' with 'u'
• Express 'dx' in terms of 'du'

Once the integral is expressed in terms of 'u', it becomes easier to evaluate. After integration, don't forget to substitute back to the original variable 'x' to complete the solution.

Indefinite Integral

An indefinite integral represents the family of all antiderivatives of a function. It is generally represented without limits and is accompanied by a constant of integration, denoted as 'C'. This constant represents the infinite number of possible antiderivatives since the derivative of a constant is zero.

Mathematically, the indefinite integral is symbolized as: \[ \int f(x)\, dx = F(x) + C \] where 'F(x)' is the antiderivative of 'f(x)'. The primary objective of finding an indefinite integral is to determine the general form of the original function before it was differentiated.

• Notation: \[ \int \]
• Always includes a constant of integration 'C'
• Find the general antiderivative of the given function

Understanding the concept of indefinite integration is crucial for solving complex integration problems that involve reversing the process of differentiation.

Integral of Square Roots

Integrating expressions involving square roots can sometimes be challenging, but several strategies can be useful, including substitution or trigonometric substitution. When faced with an integral of a square root, such as \[ \int \sqrt{ax^2+bx+c}\, dx \], one common approach is to complete the square or factor the quadratic expression to bring it to a standard form, which is more manageable.

For the integral of square roots involving perfect squares, the substitution mentioned is particularly effective.

• Use substitution to simplify the square root
• Transform to a known integral if possible
• For non-perfect squares, consider trigonometric substitution

Once simplified, these integrals often resemble standard forms that can be integrated using known formulas or further substitution techniques.

Trigonometric Substitution

The method of trigonometric substitution is utilized when dealing with integrals that involve square roots of quadratic expressions, particularly those resembling the Pythagorean theorem forms: \[a^2 - x^2, a^2 + x^2,\] and \[x^2 - a^2\].

It is a form of substitution that applies trigonometric identities to simplify the integral, making it possible to evaluate expressions that would otherwise be difficult to handle. For example, for an integral containing \[\sqrt{a^2 - x^2}\], one would substitute \[x = a \sin(\theta)\] and \[dx = a \cos(\theta) d\theta\].

• Choose a trigonometric identity related to the expression
• Substitute to simplify the square root
• Integrate using the simpler trigonometric integral

After integration, it's essential to convert the trigonometric expression back to the original variable using the appropriate inverse trigonometric function or Pythagorean relationships.