Question

Evaluate the following integrals: $$ \int \frac{\mathrm{dx}}{\mathrm{x} \sqrt{2 \mathrm{ax}-\mathrm{a}^{2}}} $$

Short answer

Question: Evaluate the integral $$\int \frac{\mathrm{dx}}{\mathrm{x} \sqrt{2 \mathrm{ax}-\mathrm{a}^{2}}}$$. Answer: $$\frac{2}{a}\sec^{-1}{\sqrt{\frac{2x}{a}-1}}+C$$

Step-by-step solution

Choose the substitution

We will make the substitution: $$ x = \frac{a}{2}(\sec^2{u} - 1) $$

Find dx

Differentiate the substitution equation with respect to u to find dx: $$ \frac{dx}{du} = a\sec^2 {u} \tan {u} $$ Then, solve for dx: $$ dx = a\sec^2 {u} \tan {u} \, du $$

Substitute x and dx into the integral

Replace x in the expression for integral and replace dx with the expression we just found: $$ \int \frac{a\;\mathrm{sec}^2{ u} \;\mathrm{tan}{ u}\;\mathrm{du}}{(\frac{a}{2}(\mathrm{sec}^2{ u} - 1) )\sqrt{2(\mathrm{a}\cdot \frac{\mathrm{a}}{2}(sec^2{ u} - 1))-(\mathrm{a}^{2})}} $$

Simplify the integral

After substituting, simplify the integrand: $$ \int \frac{a\sec^2{u}\tan{u}\, du}{(\frac{1}{2}a(\sec^2{u} - 1))\sqrt{1-\sec^2{u}}} $$ As sec^2{u} = 1 + tan^2{u}, we can write $$ \int \frac{a\sec^2{u}\tan{u}\, du}{(\frac{1}{2}a(\sec^2{u} - 1))\sqrt{1-\sec^2{u}}} = \int \frac{a(\tan^2{u} + 1)\tan{u}\, du}{(\frac{1}{2}a(\tan^2{u} + 1 - 1))\tan{u}} $$ Now cancel out the common terms: $$ \int \frac{2\, du}{a} $$

Solve the simplified integral

Integrating with respect to u, we get: $$ \frac{2}{a}\int{1\, du} = \frac{2}{a}u + C = \frac{2}{a}\sec^{-1}{\sqrt{\frac{2x}{a}-1}}+C $$ Thus, the integral evaluates to: $$ \int \frac{\mathrm{dx}}{\mathrm{x} \sqrt{2 \mathrm{ax}-\mathrm{a}^{2}}} = \frac{2}{a}\sec^{-1}{\sqrt{\frac{2x}{a}-1}}+C $$

Key concepts

Substitution Method

The substitution method is a powerful technique in integral calculus that involves changing the variable of integration to simplify an integral. It's akin to substituting a difficult question with an easier one that you can answer more readily.

When using the substitution method, you look for a part of the integral that can be replaced with a single variable, often denoted as 'u'. This is not just any random part of the integral; it should be chosen such that its differential, 'du', is also present in the integral, or can be easily obtained through manipulation. By making this substitution, the integral may be rewritten in terms of 'u', which often results in a simpler form that is easier to evaluate.

For example, in the original integral of this exercise, a substitution was made to transform the integrand into a function of a trigonometric expression. After this substitution, the integral became much simpler to handle. This is a classic move when dealing with integrals of radical functions or those involving complicated expressions.

Integrals of Trigonometric Functions

The integrals of trigonometric functions are a set of standard integrals that are frequently encountered in calculus. When functions involve trigonometric terms like sine, cosine, tangent, or their reciprocals, knowing the integrals of these basic functions is incredibly useful.

In our exercise, we encounter the secant function. We utilize the identity \(\text{sec}^2{u} = 1 + \text{tan}^2{u}\), which allows us to transform the integral into something more manageable. These trigonometric identities are essential tools in the integration process because they can convert complex expressions into simpler ones that you can integrate directly or with little additional manipulation.

To make integration easier, it’s important to be familiar with the fundamental integrals of trigonometric functions, and don't forget to consider their derivatives as well because they often appear in the integrand after substitution.

Indefinite Integration

Indefinite integration, also known as antiderivation, is the process of finding the antiderivative or the original function whose derivative yields the function being integrated. This is distinct from definite integration, which calculates the area under the curve and produces a numerical result.

When performing an indefinite integration, we are essentially reversing the differentiation process. The result is always a family of functions plus a constant of integration, denoted by 'C', because the derivative of constant is zero, any constant can be added to the antiderivative without affecting the correctness of the solution.

In the given exercise, after simplifying using trigonometric identities and substitution, the integral is reduced to the integral of one with respect to 'u'. This is a straightforward integration that results in 'u' plus the constant of integration. Afterward, it is transformed back into a function of 'x' to complete the problem. Remember, with indefinite integrals, there's always an element of reversing steps you've made, like after using the substitution method, bringing your solution back to the original variable.