Problem 35 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

Bestinmer SiC $$ \int \frac{x}{\sqrt{\frac{1}{3} x^{2}-2 x}} d x $$

Step-by-Step Solution

Answer

Use trigonometric substitution to solve and simplify integral.

Step 1: Simplify the Expression Under the Square Root

The expression under the square root is \( \frac{1}{3}x^2 - 2x \). To start simplifying, factor out the coefficient \( \frac{1}{3} \) from \( x^2 \). This gives us: \( \frac{1}{3}(x^2 - 6x) \).

Step 2: Complete the Square

The expression \( x^2 - 6x \) can be rewritten by completing the square. Add and subtract \( 3^2 \) (since \, \((6/2)^2 = 9\) \,) to get: \( x^2 - 6x + 9 - 9 \). This can be written as \((x-3)^2 - 9\).

Step 3: Rewrite the Entire Expression

Now replace \( x^2 - 6x \) in the original square root with \((x-3)^2 - 9\): \[ \int \frac{x}{\sqrt{\frac{1}{3}((x-3)^2 - 9)}} \, dx \]. Factor \( \frac{1}{3} \) inside the square root: \[ \int \frac{x}{\sqrt{\frac{1}{3}(x-3)^2 - 3}} \, dx \].

Step 4: Simplify the Integral Using Trigonometric Substitution

Use the substitution \( u = x - 3 \), which changes the integral to:\[ \int \frac{u+3}{\sqrt{\frac{1}{3}u^2 - 3}} \, du \]. Due to the form \( \sqrt{a u^2 - b} \), start using substitution.Let \( u = \sqrt{\frac{3}{a}} \sec{\theta} \) with \( a = \frac{1}{3} \) and \( b = 3 \). This simplifies to \( x = 3\sec{\theta} + 3 \). Substitute and simplify further.

Step 5: Evaluate the Integral

This substitution will eventually reduce to a standard integral form. Calculate the integral using these trigonometric identities after substitution, then back-substitute the value of \( u \) and \( x \), and simplify the result to get the solution.

Step 6: Final Result and Simplification

After evaluating and back-substituting, the integral gives an expression in terms of \( x \) and constant terms. Simplify this if needed to arrive at the final antiderivative function.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a powerful technique for evaluating integrals, especially when dealing with expressions containing square roots that resemble the Pythagorean identities. The method involves replacing a variable with a trigonometric function to simplify the integration process.

In the given exercise, the expression under the square root involves the form \( \sqrt{\frac{1}{3}u^2 - 3} \). The clue here is to recognize that this fits well with the substitution using the secant function, i.e., \( u = \sqrt{\frac{3}{a}} \sec{\theta} \), where \( a = \frac{1}{3} \).

Implementing this substitution transforms the integral into a trigonometric expression that is often easier to manage.

• This exploits the identity \( \sec^2 \theta - 1 = \tan^2 \theta \), allowing simplification of the integration process.
• After solving the integral in terms of \( \theta \), remember to convert back to the original variable \( x \) by reversing your substitution.

Keep in mind the bounds of \( \theta \) if the integral has definite limits, and ensure that transformations keep the integral equivalent through the range.

Integral Simplification

Integral simplification involves reducing the integral expression into a more manageable form before actually integrating. This might include factoring, expanding, or making strategic substitutions. In our example, this requires a few preliminary manipulations to make the integral easier to solve.

First, factor out constants from the expression within the square root to see a clearer path to simplification. By factoring out \( \frac{1}{3} \) from \( x^2 - 6x \), we can better visualize possible substitution strategies.

A subsequent trigonometric substitution (explored in the previous section) acts as an integral part of this simplification process.

• This step emphasizes organizing the original expression to fit known integral formulas and identities.
• It also sets up a smooth transition into the steps of trigonometric substitution.

Effective simplification streamlines integration, reducing computation complexity and potential errors.

Complete the Square

Completing the square is an algebraic method used to simplify quadratic expressions, commonly appearing in calculus problems involving integration. The main goal is to rewrite the quadratic in a perfect square form, enabling easier manipulation and substitution.

To complete the square for the expression \( x^2 - 6x \) in the integral, we carefully add and subtract \((3)^2 = 9\) within it, resulting in \( (x-3)^2 - 9 \). This changes the expression into a form that’s suitable for applying the trigonometric substitution discussed earlier.

• This technique transforms the quadratic expression while preserving its original value.
• It also directly assists in recognizing patterns or trigonometric identities, making the integral approachable.

Mastering this method contributes significantly to simplifying complex integrals and is an essential skill for tackling more intricate calculus problems.