Problem 39 - A Complete Resource Book in Mathematics for JEE Main 2017

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) \(\frac{1}{6(a+1)}\left(2 x^{3 a}+3 x^{2 a}+6 x^{a}\right)^{1-\frac{1}{a}}+C\) (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

(D) none of these

Step 1: Understanding the Expression

The given integral is \( \int (x^3 a + x^2 a + x a) (2x^2 a + 3x a + 6)^1 a \, dx \). We need to understand how to integrate a polynomial times another polynomial.

Step 2: Recognizing the Correct Form

We first recognize the integral of the form \( \int u \, v \, dx \), where \( u = (x^3 a + x^2 a + x a) \) and \( v = (2x^2 a + 3x a + 6) \). However, because it's simple polynomial multiplication, it will focus on direct expansion and integration rather than using techniques like integration by parts.

Step 3: Distribute the Polynomial

Multiply out the terms: \( (x^3 a + x^2 a + x a)(2x^2 a + 3x a + 6) \), which expands into a single polynomial.

Step 4: Integrate Each Term

Integrate the resulting polynomial term by term. Each term will be of the form \( ax^n \) and integrates to \( \frac{a}{n+1}x^{n+1} \). Add \( C \) at the end for the integration constant.

Step 5: Checking the Integration Result

Match the integrated expression to the options given. Compare each part to one of the options A, B, C or D to identify which one matches correctly.

Key Concepts

Polynomial Integration

Polynomial integration is a fundamental technique in calculus used to find the antiderivative of a polynomial function. Integrating a polynomial is relatively straightforward because it involves applying the power rule for integration to each term in the polynomial separately.
For a given polynomial, the process involves:
- Taking each term of the form \( ax^n \) and applying the formula \( \int ax^n \, dx = \frac{a}{n+1}x^{n+1} + C \), where \( C \) is the integration constant.
- This means we increase the exponent of \( x \) by one, and then divide the coefficient by this new exponent.
This method makes it easy to find the antiderivative, and is particularly useful when dealing with simpler polynomial expressions.

Integral Calculus

Integral calculus is a branch of mathematics focused on the concept of integration. It plays a crucial role in finding the total size or value, such as areas under curves, accumulated quantities, and many other applications. In the context of polynomials, integral calculus provides tools to work with antiderivatives and compute definite and indefinite integrals.
When you take an integral, you are essentially reversing the process of differentiation. For polynomials, this involves finding a function whose derivative gives back the original polynomial.
The essential components include:

• Indefinite Integral: Represents a family of functions and includes a constant of integration \( C \).
• Definite Integral: Computes the net area under a curve between specific limits, without requiring \( C \).

By mastering integral calculus, one can handle a broad variety of functions, particularly polynomials, and solve real-life problems with mathematical precision.

Step-by-Step Integration

Step-by-step integration is a valuable approach, especially when dealing with complex expressions, like the product of two polynomials. This method encourages breaking down the integration process into smaller, manageable steps to enhance understanding and accuracy.
It typically involves:

• Understanding the structure of the expression, identifying whether it's a polynomial, product of functions, or something else.
• Expanding the expression if needed, such as multiplying polynomials to create a single polynomial.
• Integrating term by term after expanding, applying integration rules to each individual term.
• Checking the integrated result against given options or verifying with potential solutions.

This methodical approach ensures that errors are minimized and allows for clear checking against possible solutions, which is especially relevant in educational contexts where learners need to firmly grasp the steps involved.