Problem 19 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\int \frac{\sqrt{x}}{\sqrt{x^{3}+4}} d x\) equals (A) \(\frac{2}{3} \ln \left(\frac{2}{\sqrt{x^{3}}-\sqrt{x^{3}-4}}\right)+C\) (B) \(\frac{2}{3} \ln \left(\frac{2}{\sqrt{x^{3}}-\sqrt{x^{3}-4}}\right)+C\) (C) \(\frac{2}{3} \ln \left(\frac{2}{\sqrt{x^{3}}-\sqrt{x^{3}-4}}\right)+C\) (D) none of these

Step-by-Step Solution

Answer

The answer is D: none of these options are correct.

Step 1: Choose a Substitution

We will use the substitution method to solve this integral. Let us choose \( u = \sqrt{x^3 + 4} \). So, \( u^2 = x^3 + 4 \) and differentiate both sides to get \( 2u \frac{du}{dx} = 3x^2 \).

Step 2: Express dx in terms of du and x

From \( 2u \frac{du}{dx} = 3x^2 \), we can rearrange to find \( dx \). We have: \( \frac{du}{dx} = \frac{3x^2}{2u} \), hence \( dx = \frac{2u}{3x^2} \, du \).

Step 3: Replace Variables in the Integral

In the integral, replace \( \sqrt{x} \) with \( x^{1/2} \), and from \( x^3 = u^2 - 4 \), we find \( x = (u^2 - 4)^{1/3} \). Substitute these into the integral: \[ \int \frac{x^{1/2}}{u} \cdot \frac{2u}{3x^2} \, du = \int \frac{2 x^{1/2}}{3x^2} \, du \].

Step 4: Simplify the Expression

Simplify \( \frac{x^{1/2}}{x^{2}} = x^{-3/2} \) obtained from the substitution: \[ \int \frac{2}{3} x^{-3/2} \, du \]. Given \( x = (u^2 - 4)^{1/3} \), calculate \( x^{-3/2} = [(u^2 - 4)^{1/3}]^{-3/2} = (u^2 - 4)^{-1/2} \). Substituting into the integral simplifies it further: \[ \int \frac{2}{3} (u^2 - 4)^{-1/2}\, du\].

Step 5: Solve the Integral

Use a standard integral table to solve \( \int (u^2 - 4)^{-1/2} \, du \), which results in \( \ln|u + \sqrt{u^2 - 4}| \). Thus, the integral transforms to: \[ \frac{2}{3} \ln|u + \sqrt{u^2 - 4}| + C \].

Step 6: Reverse Substitution

Return to the original variables using \( u = \sqrt{x^3 + 4} \). Substitute back to get \[ \frac{2}{3} \ln| \sqrt{x^3 + 4} + \sqrt{x^3}| + C \]. This expression is equivalent to option D, which states it's none of these because none are an exact match to our derived integral.

Key Concepts

Substitution Method

The substitution method is a helpful technique for solving integrals in calculus. By using this method, we make the integral simpler by transforming it into a potentially easier problem.

With substitution, we choose a part of the integral to represent with a new variable. For example, in our problem, we used the substitution \( u = \sqrt{x^3 + 4} \). The goal here was to express the integral in terms of \( u \) instead of \( x \). This often makes the integral easier to solve. The new variable \( u \) helps eliminate complicated terms in the original integral.

• First, identify a substitution that simplifies the expression. Here, \( u^2 = x^3 + 4 \).
• Differentiate to find \( du \). In our example, this gave us \( 2u \frac{du}{dx} = 3x^2 \).
• Solve for \( dx \) in terms of \( du \), which we found to be \( dx = \frac{2u}{3x^2} du \).

By making these transformations, you can tackle otherwise challenging integrals with a new perspective. Substitution often makes what's confusing become clear.

Definite Integrals

Definite integrals represent the area under a curve between two points. Although we worked on an indefinite integral in this exercise, understanding definite integrals is important too.

A definite integral is written like this: \( \int_{a}^{b} f(x) \, dx \). It has boundaries, \( a \) and \( b \), and it calculates the net area between these bounds.

• The integral is evaluated by finding the antiderivative \( F(x) \) of \( f(x) \).
• Then, substitute \( b \) and \( a \) into \( F(x) \), and compute: \( F(b) - F(a) \).

Definite integrals provide a numerical description of the total accumulation, such as distance traveled over time or the total area between a curve and the x-axis. While the given exercise was an indefinite integral, the skills learned apply to definite integrals too.

Indefinite Integrals

Indefinite integrals, like the one in our exercise, do not involve specific bounds. Unlike definite integrals, they result in a function plus a constant, \( C \), and can represent many anti-derivatives.

The purpose of solving indefinite integrals is to find a general form of the antiderivative for a function. The example \( \int \frac{\sqrt{x}}{\sqrt{x^3+4}} dx \) illustrates this process. Our [solution involved finding a general form and included a constant] of integration \( C \).

• Think of the integral \( \int f(x) \, dx \) as the reverse of differentiation.
• The integration constant \( C \) is crucial because it indicates that many functions could satisfy the integral.

Indefinite integration helps build a toolkit for solving differential equations and understanding changes within different contexts. It provides the foundational piece used before applying initial conditions or bounds in definite integrals.