Problem 39 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\int\left(x^{3} a+x^{2} a+x a\right)\left(2 x^{2} a+3 x a+6\right)^{1} a d x=\) (A) \(\frac{1}{6(a+1)}\left(2 x^{3 a}+3 x^{2 a}+6 x^{a}\right)^{1-\frac{1}{4}}+C\) (B) \(\frac{1}{6(a+1)}\left(2 x^{3 a}+3 x^{2 a}+6 x^{a}\right)^{1+\frac{1}{a}}+C\) (C) \(\frac{1}{3(a+1)}\left(2 x^{3 a}+3 x^{2 a}+6 x^{a}\right)^{1+\frac{1}{a}}+C\) (D) none of these

Step-by-Step Solution

Answer

The answer is (D) none of these.

Step 1: Simplify the Integral Expression

We start by writing the integral expression: \[ \int\left(x^{3} a + x^{2} a + x a\right)\left(2 x^{2} a + 3 x a + 6\right)a dx \] We can simplify the expression by distributing: \[ a \int\left(x^{3} a + x^{2} a + x a\right)\left(2 x^{2} a + 3 x a + 6\right) dx \]The factor \( a \) can be factored out of the integral as it is a constant with respect to \( x \): \[ a \int\left(x^{3} a + x^{2} a + x a\right)\left(2 x^{2} a + 3 x a + 6\right) dx \]

Step 2: Review the Expression in Terms of Polynomial Expansion

Let's consider expanding the polynomial as if we were setting up an expression of the form \(u = \left(2 x^{2} a + 3 x a + 6\right)\). Notice that the presence of these terms will favor substitutions later. At this stage, we consider simplifying using polynomial expansion or checking for a pattern that fits a direct identification.

Step 3: Identify Suitable Substitution

We rewrite the structure of the expression suggesting substitution of variables. Assume \( u = (2 x^{2} a + 3 x a + 6) \) with which our differential components can align as terms multiply naturally when differentiated.

Step 4: Calculate the Derivative of the Polynomial Expression

We calculate the derivative of \( u \) with respect to \( x \): \[ du = (4 x a^2 + 3 a) dx \].

Step 5: Setup Integration Using Substitution

Using \( u \) substitution, we substitute back in to set the form of the integration: \[ a \int u^1 \cdot f'(x) dx \]This encourages a potential simplification, noting that we might need to change index or perform further factoring.

Step 6: Choose Matching Solution Expression

Since our setup doesn't fit straightforwardly any transformation that simplifies into the solution form directly while ensuring correctness, confirm solution matches none as expected correct structure didn't arise within expression syntax solved down.

Key Concepts

Polynomial Expansion

Polynomial expansion is a technique used to simplify expressions that involve polynomials. When you have a polynomial expression, the goal is to spread or expand it in such a way that makes the calculation easier. This technique involves breaking down larger expressions into smaller terms or factors. It is particularly useful when you need to integrate complex polynomial expressions as it can help to see potential simplifications or patterns that might not be immediately obvious.

To perform polynomial expansion, you essentially distribute each term across the others, much like the distributive property you might have learned in basic algebra. For the integral \[ \int\left(x^{3} a + x^{2} a + x a\right)\left(2 x^{2} a + 3 x a + 6\right)a dx \] you would look at each component, expand them, and multiply the terms across each other. This allows you to see related terms that can be combined or simplified before integrating.

It's important to understand that polynomial expansion is crucial in identifying patterns or structures in expressions that lend themselves to certain integration techniques like substitution.

Substitution Method

The substitution method is a fundamental technique in calculus used to transform a complex integral into a simpler form. It involves substituting a part of the integral with a new variable to simplify the integration process.

For example, in our original exercise, we let \( u = (2 x^{2} a + 3 x a + 6) \). This substitution signifies changing the variable of integration from \( x \) to \( u \), which can make the integral easier to solve. Along with substitution, you must deal with the differential \( dx \) and match it with \( du \), the derivative term. In our case, we arrived at \( du = (4x a^2 + 3a) dx \) when calculating the derivative.

The substitution method often pairs with polynomial expansion because once expanded, the terms of the polynomial can indicate which substitutions may simplify them into a recognizable form. After substitution, the integral bounds or limits if defined, or each term in the expression, shift from terms of \( x \) to terms of \( u \). Sometimes substituting back after solving with respect to \( u \) is necessary to express the final results in terms of the original variable.

Differential Calculus

Differential calculus is the branch of calculus that deals with the study of rates at which quantities change. It involves differentiation, which is the process of finding a derivative. The derivative measures how a function changes as its input changes.

In the context of our exercise, finding the derivative of the polynomial function \( u = (2 x^{2} a + 3 x a + 6) \) was an integral step for the substitution method. Derivatives are crucial because they help in determining the rate of change of the function, which is essential for integration, especially when using the substitution method.

To calculate the derivative of a polynomial expression like \( u \), you apply rules of differentiation like the power rule. For a simple term like \( x^n \), the derivative is \( n x^{n-1} \). This allows you to compute \( du \), which becomes the differential component in the integral function. This technique shows how closely linked the processes of differentiation and integration are, underscoring the importance of mastering differential calculus for tackling integration problems effectively.