Problem 70 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\int \frac{d x}{(1+\sqrt{x})^{8}}=\frac{-1}{3(1+\sqrt{x})^{k_{1}}}+\frac{2}{7(1+\sqrt{x})^{k_{2}}}+C\) where (A) \(k_{1}=6\) (B) \(k_{2}=7\) (C) \(k_{1}=-6\) (D) \(k_{2}=-7\)

Step-by-Step Solution

Answer

(A) \(k_{1}=6\) and (B) \(k_{2}=7\) are correct.

Step 1: Understand the Problem

We are given an integral \( \int \frac{dx}{(1+\sqrt{x})^8} \) and we need to confirm which values of \( k_1 \) and \( k_2 \) satisfy the solution given as \( \frac{-1}{3(1+\sqrt{x})^{k_1}}+\frac{2}{7(1+\sqrt{x})^{k_2}}+C \).

Step 2: Simplify the Integral

We introduce the substitution \( u = 1 + \sqrt{x} \), so \( du = \frac{1}{2\sqrt{x}}dx \) or equivalently, \( 2(u - 1)du = dx \). The integral becomes \( \int \frac{2(u-1)du}{u^8} = \int 2\left(\frac{1}{u^7} - \frac{1}{u^8}\right)du \).

Step 3: Integrate Each Term Separately

Integrate each term separately: 1. \( \int \frac{2}{u^7} du = 2\int u^{-7} du = -\frac{2}{6u^6} = -\frac{1}{3u^6} \) 2. \( \int \frac{-2}{u^8} du = -2\int u^{-8} du = \frac{2}{7u^7} \)

Step 4: Combine and Compare to Problem Statement

The combined result is \( -\frac{1}{3u^6} + \frac{2}{7u^7} + C \), which matches the given expression \( \frac{-1}{3(1+\sqrt{x})^{k_1}} + \frac{2}{7(1+\sqrt{x})^{k_2}} + C \) where \( u = 1 + \sqrt{x} \).

Step 5: Determine Values for \( k_1 \) and \( k_2 \)

From the expression \( \frac{-1}{3u^6} + \frac{2}{7u^7} \), we can directly see \( k_1 = 6 \) and \( k_2 = 7 \).

Step 6: Conclusion

The correct values satisfying the conditions are \( k_1 = 6 \) and \( k_2 = 7 \). Thus, options (A) and (B) are correct.

Key Concepts

Substitution Method

The substitution method is a valuable tool in integral calculus, making it easier to solve complex integrals by simplifying the integral's form. In our problem, we used the substitution \( u = 1 + \sqrt{x} \). By doing so, we transformed our original integral \( \int \frac{dx}{(1+\sqrt{x})^8} \) into a more manageable form.The idea is to replace a part of the integrand with a new variable \( u \) so that the integral becomes simpler. Once you make this substitution, you'll also need to express \( dx \) in terms of \( du \). For our case, differentiating both sides of the substitution gives us \( du = \frac{1}{2\sqrt{x}}dx \). From there, we rearrange to find \( dx = 2(u - 1)du \).By substituting these into the original integral, we simplify it to \( \int \frac{2(u-1)du}{u^8} \). This method essentially redefines the integral's frame of reference, making it easier to evaluate.Remember the substitution method is effective when there is a clear substitution that simplifies the integral. It's commonly used for integrals involving compositions of functions or when the integrand has a specific structure that matches a derivative.

Integral Calculus

Integral calculus is a branch of mathematics focusing on integrals and their properties. This concept was key to solving our given problem.For this exercise, integral calculus was employed to evaluate the transformed integrals: \( \int \frac{2}{u^7} du \) and \( \int \frac{-2}{u^8} du \). Through integral calculus, we determine the antiderivatives:

• \( \int 2u^{-7} du = -\frac{2}{6}u^{-6} = -\frac{1}{3u^6} \)
• \( \int -2u^{-8} du = \frac{2}{7}u^{-7} \)

These processes show the power of integral calculus in breaking down complex expressions into simpler form. Integral calculus allows us to understand areas under curves, compute limits, and solve real-world problems involving accumulation.It’s important to note that integral calculus not only involves finding antiderivatives but also understanding the fundamental theorem of calculus, which links differentiation and integration, providing insights into evaluating definite integrals.

Mathematical Problem Solving

Mathematical problem solving is a critical skill in calculus and involves applying different methods and tools to arrive at a solution. In the context of our exercise, problem solving was crucial.Initially, understanding the format of the original integral \( \int \frac{dx}{(1+\sqrt{x})^8} \) guided the choice of method. Identifying that substitution would simplify the integral was the key first step. Subsequently, evaluating the transformed integral required knowledge of integral calculus for solving each component.Step-by-step thinking helped in distinguishing which transformations and integrations would produce the desired result. Finally, comparing the antiderivative with the original problem’s structure allowed us to verify and find the correct values of \( k_1 \) and \( k_2 \).Mathematical problem solving often requires:

• Identifying problem features and patterns
• Choosing the right mathematical tool or method
• Applying logical reasoning to find solutions
• Verifying results by comparing them with the problem's requirements

By integrating these practices, problem-solving becomes a logical sequence of actions leading to a clear and correct solution.