Problem 64 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\int \frac{d x}{\left(2 a x+x^{2}\right)^{3 / 2}}=\) (A) \(\frac{1}{a^{2}} \frac{x+a}{\sqrt{x^{2}+2 a x}}+c\) (B) \(\frac{1}{a^{2}} \frac{x-a}{\sqrt{x^{2}+2 a x}}+c\) (C) \(-\frac{1}{a^{2}} \cdot \frac{x+a}{\sqrt{x^{2}+2 a x}}+c\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (C).

Step 1: Substitution Introduction

To solve the integral \(\int \frac{dx}{(2ax + x^2)^{3/2}}\), we look for a suitable substitution to simplify the expression under the square root. Notice that the expression \(2ax + x^2\) can be factored as \((x+a)^2 - a^2\). It suggests we might use the substitution \(x+a = a \tan(\theta)\).

Step 2: Apply Substitution

Using the substitution \(x+a = a \tan(\theta)\), we have \(dx = a \sec^2(\theta) \, d\theta\). Also, \(x^2 + 2ax = (a \tan(\theta))^2 - a^2 = a^2 \sec^2(\theta) - a^2 \). The integral becomes:\[ \int \frac{a \sec^2(\theta) \, d\theta}{(a^2 \sec^2(\theta) - a^2)^{3/2}}.\]

Step 3: Simplify the Integral

The expression \(a^2 \sec^2(\theta) - a^2\) simplifies to \(a^2 \tan^2(\theta)\). Thus, the integral becomes:\[ \int \frac{a \sec^2(\theta) \, d\theta}{a^3 \tan^3(\theta)} = \frac{1}{a^2} \int \frac{ \sec^2(\theta)}{\tan^3(\theta)} \, d\theta. \] This simplifies further to \(\frac{1}{a^2} \int \cot^2(\theta) \, d\theta.\)

Step 4: Solve the Simplified Integral

We solve \( \int \cot^2(\theta) \, d\theta\). Using the identity \(\cot^2(\theta) = \csc^2(\theta) - 1\), we get:\[ \int \cot^2(\theta) \, d\theta = \int (\csc^2(\theta) - 1) \, d\theta = -\cot(\theta) - \theta + C. \]

Step 5: Back Substitution

We revert back to the variable \(x\) using our initial substitution. Since \(x+a = a \tan(\theta)\), we have \( \tan(\theta) = \frac{x+a}{a} \). Also, \(\cot(\theta) = \frac{a}{x+a}\). For \(\theta\), using trigonometric identities, we find \(\theta = \arctan\left(\frac{x+a}{a}\right)\). The antiderivative is thus:\[ -\frac{x+a}{a} - \arctan\left(\frac{x+a}{a}\right) + c. \]

Step 6: Match the Answer

Rewriting the antiderivative, it resembles answer (C): \(-\frac{1}{a^2} \cdot \frac{x+a}{\sqrt{x^2+2ax}}+C\), since \(\arctan\left(\frac{x+a}{a}\right)\) can be absorbed into the constant \(C\). Thus, the answer closely matches option (C).

Key Concepts

Integration Techniques

Integration is a powerful tool in calculus used to find areas, volumes, central points, and much more. There are several techniques available for solving integrals, each being effective in different scenarios. Choosing the right approach can simplify complex-looking expressions.

• Substitution Method: This is where you replace a complicated part of the integrand with a new variable to make the integration process easier. For example, in the exercise, the expression \(2ax + x^2\) is substituted by \((x+a)^2 - a^2\), simplifying the integration process.
• Integration by Parts: This technique is often used when the integrand is a product of two functions. It is based on the product rule for differentiation.
• Trigonometric Substitution: This involves using trigonometric identities to simplify integrals, usually when dealing with square roots. This is relevant to the exercise, where \(x+a = a \tan(\theta)\) was used to help solve the problem.

Considering these techniques helps to solve integrals more efficiently and effectively.

Trigonometric Substitution

Trigonometric substitution is a specific method used to evaluate integrals that involve square roots of quadratic expressions. It is particularly handy in calculus problems where algebraic manipulation alone is insufficient.In this exercise, we used trigonometric substitution to solve the integral by recognizing the expression under the square root, \(2ax + x^2\), as something that can be rewritten using trigonometric identities.

• When you notice a term like \((x^2 + 2ax)\), rephrasing it in terms of a trigonometric function can offer a new perspective. Here, it was rewritten in a form compatible with the Pythagorean identity.
• The substitution \(x+a = a \tan(\theta)\) was introduced. This transformed the problem into a trigonometric integral, making it easier to solve.
• The key benefit of this approach is reducing the complexity of the integral, by taking advantage of basic trigonometric identities like \(\sec^2(\theta) = 1 + \tan^2(\theta)\).

Using this method effectively requires a solid understanding of trigonometric identities and how they relate to calculus.

Indefinite Integrals

Indefinite integrals represent the family of all antiderivatives of a function. Unlike definite integrals, they do not evaluate to a numerical value but instead produce a general formula.

• An indefinite integral is represented as \(\int f(x) \, dx = F(x) + C\), where \(F(x)\) is the antiderivative and \(C\) is the constant of integration.
• The presence of \(C\) indicates that the antiderivative is not unique and can differ by a constant, acknowledging all the possible shifts of the function graph vertically.
• While solving the definite integral problem in the exercise, we ultimately derived an indefinite integral form. This means we discovered a general solution which solves the integral without upper and lower limits.

Indefinite integrals are crucial in solving differential equations and determining the general behavior of functions in calculus.