Problem 84 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\int \sqrt[3]{x} \sqrt[7]{1+\sqrt[3]{x^{4}}} d x\) is equal to (A) \(\frac{21}{32}\left(1+\sqrt[3]{x^{4}}\right)^{8 / 7}+C\) (B) \(\frac{32}{21}\left(1+\sqrt[3]{x^{4}}\right)^{8 / 7}+C\) (C) \(\frac{7}{32}\left(1+\sqrt[3]{x^{4}}\right)^{8 / 7}+C\) (D) none of these

Step-by-Step Solution

Answer

Option (A): \( \frac{21}{32}(1+\sqrt[3]{x^{4}})^{8/7} + C \) is correct.

Step 1: Substitution Setup

To simplify the integral, start by making a substitution. Let's set \( u = 1 + \sqrt[3]{x^4} \). Differentiate both sides to find an expression for \( du \). This gives us \( du = \frac{4}{3} x^{1/3} \, dx \). Thus, \( x^{1/3} \, dx = \frac{3}{4} \, du \).

Step 2: Rewrite the Integral

Substitute \( u \) and \( du \) into the integral. The original integral becomes \( \int x^{1/3} \sqrt[7]{1 + x^{4/3}} \, dx = \int \frac{3}{4} u^{1/7} \, du \) after substitution.

Step 3: Integration

Now integrate \( \int \frac{3}{4} u^{1/7} \, du \). This can be integrated using the power rule: \( \frac{3}{4} \cdot \frac{u^{1/7 + 1}}{1/7 + 1} \). Simplify the exponent: \( 1/7 + 1 = 8/7 \), and thus the integral becomes \( \frac{3}{4} \cdot \frac{7}{8} u^{8/7} + C \).

Step 4: Simplify the Expression

The expression simplifies to \( \frac{21}{32} u^{8/7} + C \), where \( u = 1 + \sqrt[3]{x^4} \). Thus, substituting back for \( u \), the solution is \( \frac{21}{32} (1 + \sqrt[3]{x^4})^{8/7} + C \).

Step 5: Select the Correct Option

Compare the simplified integral solution to the options provided. The solution \( \frac{21}{32}(1+\sqrt[3]{x^{4}})^{8/7} + C \) matches option (A).

Key Concepts

Substitution Method

The Substitution Method in integration is a powerful tool to simplify complex integrals, especially those involving composite functions. This approach works by substituting a part of the integral with a new variable to transform it into a simpler form.

In our exercise, the substitution is key to simplifying the integral \(\int \sqrt[3]{x} \sqrt[7]{1+\sqrt[3]{x^{4}}} \, dx\). Here, we set \(u = 1 + \sqrt[3]{x^{4}}\). This clever substitution targets the composite part \(\sqrt[7]{1+\sqrt[3]{x^{4}}}\), making the integration more straightforward.

To proceed, differentiate \(u\) with respect to \(x\) to find an expression for \(du\). This gives \(du = \frac{4}{3} x^{1/3} \, dx\), which implies \(x^{1/3} \, dx = \frac{3}{4} \, du\). With this substitution, the original integral becomes significantly simpler.

• Identify the composite part of the function.
• Choose a substitution that simplifies this part.
• Convert the entire integral in terms of the new variable.
• Solve the transformed integral using standard methods.

By substituting, not only do we simplify the integral but also align it with techniques like the Power Rule, making it easier to solve.

Integration by Parts

Integration by Parts is a technique based on the product rule of differentiation, used when integrating products of functions. However, it's not directly applicable in our exercise. Still, understanding this method provides deeper insights into integration techniques.

The identity for Integration by Parts is:\[\int u \, dv = uv - \int v \, du\] To use Integration by Parts:

• Select functions \(u\) and \(dv\) from your integral.
• Differentiate \(u\) to get \(du\), and integrate \(dv\) to find \(v\).
• Substitute into the formula to solve the integral.

While our problem uses substitution and not Integration by Parts, knowing this method prepares you for problems where products can't be simplified easily through substitution alone. It teaches a structured approach to decomposing functions into simpler fragments.

Power Rule in Integration

The Power Rule in Integration is a fundamental technique used for integrating functions of the form \(x^n\). This rule transforms the function into a more straightforward expression by raising the power by one and dividing by the new power.

The rule states:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\] This is applied to functions where \(n eq -1\). In the exercise, after substitution leads to \(\int \frac{3}{4} u^{1/7} \, du\), we apply the Power Rule to find the integral.

For \(u^{1/7}\), increase the exponent by 1 to get \(u^{8/7}\) and divide by this new exponent:\[\frac{3}{4} \cdot \frac{u^{8/7}}{8/7} + C = \frac{3}{4} \cdot \frac{7}{8} u^{8/7} + C\]Simplifying gives \(\frac{21}{32} u^{8/7} + C\).

• Identify the power of \(x\) or \(u\) in your integral.
• Apply the Power Rule to integrate.
• Simplify the result.

Understanding and using the Power Rule allows for quicker integration of polynomials and expressions that can be rewritten with power terms, as seen here where substitution paired with the Power Rule leads directly to the solution.