Problem 83 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\int \frac{\sqrt[3]{1+\sqrt[4]{x}}}{\sqrt{x}} d x\) is equal to (A) \(12\left(\frac{(1+\sqrt[4]{x})^{n / 3}}{7}+\frac{(1+\sqrt[4]{x})^{4 / 3}}{4}\right)+C\) (B) \(12\left(\frac{(1+\sqrt[4]{x})^{7 / 3}}{7}-\frac{(1+\sqrt[4]{x})^{4 / 3}}{4}\right)+C\) (C) \(6\left(\frac{(1+\sqrt[4]{x})^{7 / 3}}{7}-\frac{(1+\sqrt[4]{x})^{4 / 3}}{4}\right)+C\) (D) none of these

Step-by-Step Solution

Answer

The answer is (B).

Step 1: Simplify the Integral

First, let's rewrite the given integral: \( \int \frac{\sqrt[3]{1+\sqrt[4]{x}}}{\sqrt{x}} d x \). We need to find simpler expression by substituting. Let \( u = 1+\sqrt[4]{x} \), meaning \( \sqrt[4]{x} = u - 1 \), so \( x = (u-1)^4 \). We also have to express \( dx \) in terms of \( du \).

Step 2: Differentiate the Substitution

Differentiate the substitution equation \( u = 1+\sqrt[4]{x} \). Thus, \( \frac{du}{dx} = \frac{1}{4x^{3/4}} \) or \( dx = 4(u-1)^3 du \).

Step 3: Substitute and Simplify

Substitute \( dx = 4(u-1)^3 du \) and \( \sqrt{x} = \sqrt{(u-1)^4} = (u-1)^2 \) into the integral. This gives: \[ \int \frac{\sqrt[3]{u}}{(u-1)^2} \cdot 4(u-1)^3 du = 4 \int u^{1/3} (u-1) du \].

Step 4: Expand and Integrate

Next, expand the integrand: \( 4 \int (u^{1/3} u - u^{1/3}) du = 4 \int (u^{4/3} - u^{1/3}) du \). Integrate each term: \( 4 \left[ \frac{u^{7/3}}{7/3} - \frac{u^{4/3}}{4/3} \right] + C \).

Step 5: Simplify the Result

Simplify the integrated terms: \( 4 \left( \frac{3}{7} u^{7/3} - \frac{3}{4} u^{4/3} \right) + C = 12\left( \frac{u^{7/3}}{7} - \frac{u^{4/3}}{4} \right) + C \).

Step 6: Substitute Back

Finally, substitute back \( u = 1 + \sqrt[4]{x} \) into the expression: \( 12\left( \frac{(1+\sqrt[4]{x})^{7/3}}{7} - \frac{(1+\sqrt[4]{x})^{4/3}}{4} \right) + C \). This matches option (B).

Key Concepts

Integration Techniques

Integration is a fundamental concept in calculus, which often involves finding the area under a curve. In integral calculus, different techniques are used to tackle complex integrals and make the process of finding antiderivatives simpler. One well-known technique is substitution, which can be particularly effective when dealing with integrals that contain composite functions.

Substitution transforms the variable of integration, simplifying the integral's appearance and making it easier to compute. Another trick is algebraic simplification, where the goal is to rewrite the integral in a form that is easier to deal with. This involves using algebraic identities and properties of exponents and roots.

By combining these integration techniques, students can solve various integrals that appear daunting at first sight. These methods offer a strategic approach, allowing one to break down the problem into more manageable steps.

Substitution Method

The substitution method is a powerful tool in integral calculus for simplifying and solving integrals. It's particularly handy when an integral includes a composite function. The idea is to substitute part of the integral with a new variable. This simplifies the integral and makes finding the antiderivative more feasible.

In our exercise:

• We defined a new variable, let's say, by setting \( u = 1 + \sqrt[4]{x} \). This substitution offers a cleaner form of the integral.
• The substitution translates \( dx \) in terms of \( du \). Differentiating the substitution equation gives a way to replace the original differential \( dx \) with \( du \), which simplifies the integral further.
• The new variable \( u \) makes the integral easier to integrate, simplifying the algebra and exponent manipulation.

The substitution method transforms the integral into a standard form that is much easier to handle, resulting in a straightforward integration process.

Algebraic Simplification

Algebraic simplification plays a crucial role in breaking down complex integrals into simpler forms that can be tackled more easily. An integral given in a complicated form can often be made simpler by manipulating its algebraic expressions.

For example, in the exercise provided, part of the process involved simplifying expressions like \( \frac{\sqrt[3]{1 + \sqrt[4]{x}}}{\sqrt{x}} \). This requires understanding properties of exponents and radicals:

• Manipulating the expression into a form where substitutions are more evident.
• Reducing complex fractions and radicals using algebraic identities and basic rules of exponents.

Once simplified, the integration process becomes much more straightforward. Often this simplification is applied right before substitution or during it, to ensure the simplest form is achieved before the integral is worked out completely. Thus, algebraic simplification makes handling integrals less daunting and more systematic.