Problem 229 - A Complete Resource Book in Mathematics for JEE Main

Question

The integral \(\int_{2}^{4} \frac{\log x^{2}}{\log x^{2}+\log \left(36-12 x+x^{2}\right)} d x\) is equal to: (A) 4 (B) 1 (C) 6 (D) 2

Step-by-Step Solution

Answer

The integral evaluates to 2, option (D).

Step 1: Simplify the Integral Expression

Let's consider the function inside the integral: \( \frac{\log x^{2}}{\log x^{2}+\log \left(36-12x+x^{2}\right)} \). We can use the property of logarithms that states \( \log(a) + \log(b) = \log(ab) \). The expression thus simplifies to \( \frac{\log x^{2}}{\log(x^{2}(36-12x+x^{2}))} \).

Step 2: Rewrite as a Symmetric Integral

Notice that \(36 - 12x + x^{2}\) can be rewritten as \((x-6)^2\). Now, we will simplify the integral by looking for symmetry. The integral becomes \( \int_{2}^{4} \frac{\log x^{2}}{\log\left((x^{2}((x-6)^2))\right)} \, dx \). Recognizing a possible symmetry with \( x \to 6 - x \), let's change variable to see its effect.

Step 3: Use Symmetry and Substitution

Let \( u = 6-x \), so that when \( x = 2, u = 4 \) and \( x = 4, u = 2 \). This suggests symmetry. Both bounds switch roles under this substitution, and the integral becomes: \[ \int_{4}^{2} \frac{\log(6-u)^{2}}{\log\left((6-u)^{2}\right) + \log\left(36 - 12(6-u) + (6-u)^{2}\right)} \, (-du). \] After simplification, this is the same as reversing the integral \( \int_{2}^{4} \frac{\log x^{2}}{\log(x^{2} + x^{2}(6-u)^2)} \, dx \).

Step 4: Conclusion Due to Symmetry

Due to symmetry and substitution, you find the function inside the integral evaluates as 1 at each corresponding \( x \) and \( 6-x \). That means the integral of a constant 1 over \([2, 4]\) with respect to \( x \) simply results in the length of the interval. The interval \([2, 4]\) has a length of \(4 - 2 = 2\). Therefore, the integral evaluates to 2.

Key Concepts

Logarithmic Integration

Logarithmic integration involves working with functions that contain logarithms. Logarithm properties can be very helpful in simplifying expressions. In our original exercise, we start with the expression \( \frac{\log x^{2}}{\log x^{2} + \log(36 - 12x + x^{2})} \). One key property is \( \log(a) + \log(b) = \log(ab) \), which combines logarithms of multiplied values into one. Applying this property, the denominator becomes \( \log(x^{2}(36 - 12x + x^{2})) \). By reducing complex logarithmic expressions, students can focus on applying calculus methods like integration.
Logarithmic integration requires careful handling of the log rules and understanding their properties in order to break down integrals into simpler functions that are easier to work with.

Symmetry in Integration

Symmetry in integration is a strategy that simplifies problems by recognizing patterns. It exploits how parts of a function repeat or mirror each other. In our exercise, the polynomial \(36 - 12x + x^{2}\) can be rewritten as \((x-6)^2\). This hints at potential symmetry, as expressions with squared terms often have symmetric properties.

• We use the variable substitution \(x \to 6 - x\) to explore this symmetry.
• This substitution transforms the integral limits, swapping \([2,4]\) to \([4,2]\).

Discovering symmetry can significantly reduce the complexity of the integral by showing that two parts of the integral exactly match. Hence we find that evaluating over an interval leads to a constant, simplifying our results and making complex calculations more manageable.

Substitution Method

The substitution method is an essential tool in integration where you change variables to simplify the integration process. By choosing a new variable, often denoted as \(u\), you aim to transform the integral into an easier form. In our example, we let \(u = 6-x\), which corresponds logically to a reflection around 6.

This choice of \(u\) simplifies the integral bounds and respects the symmetry that we've previously identified. The substitution results in reversing the limits of integration, changing \(x=2\) to \(u=4\) and \(x=4\) to \(u=2\), effectively flipping the interval.

• When substituting, remember to also change \(dx\) to \(-du\).
• The expression inside the integral also changes accordingly, leading to a simplified form of the integral.

The substitution method is a versatile technique that can be combined with recognition of symmetry to drastically simplify finding the solution to an integral.