Problem 34 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\int x\left\\{f\left(x^{2}\right) g^{\prime \prime}\left(x^{2}\right)-f^{\prime \prime}\left(x^{2}\right) g\left(x^{2}\right)\right\\} d x=\) (A) \(f\left(x^{2}\right) g^{\prime}\left(x^{2}\right)-g\left(x^{2}\right) f^{\prime}\left(x^{2}\right)+c\) (B) \(\frac{1}{2}\left\\{f\left(x^{2}\right) g\left(x^{2}\right) f^{\prime}\left(x^{2}\right)\right\\}+c\) (C) \(\frac{1}{2}\left\\{f\left(x^{2}\right) g^{\prime}\left(x^{2}\right)-g\left(x^{2}\right) f^{\prime}\left(x^{2}\right)\right\\}+c\) (D) none of these

Step-by-Step Solution

Answer

Option (C) is correct.

Step 1: Identify the Structure of the Integral

The given integral is of the form \( \int x \{ f(x^2) g''(x^2) - f''(x^2) g(x^2) \} \, dx \). This resembles a known pattern related to variations of functions.

Step 2: Recognize the Application of Leibniz Rule

Recognize that the form \( f(x^2)g''(x^2) - f''(x^2)g(x^2) \) is similar to the formula involving derivatives of a product \( (uv)'' = u''v + 2u'v' + uv'' \). This indicates a pattern where integration is straightforward.

Step 3: Simplify with Substitution

Let \( t = x^2 \). Then \( dt = 2x \, dx \) or \( x \, dx = \frac{1}{2} dt \). Substitute into the integral to transform it into \( \frac{1}{2} \int \{ f(t) g''(t) - f''(t) g(t) \} \, dt \).

Step 4: Apply Integration by Parts

Utilize the integral form \( \int (u''v - uv'') \, dt \) which results in \( uv' - vu' \) up to a constant of integration. Applying this, the integral becomes \( \frac{1}{2} [f(t) g'(t) - g(t) f'(t)] + c \).

Step 5: Convert Back to Original Variables

Replace back \( t = x^2 \) to get \( \frac{1}{2} [f(x^2) g'(x^2) - g(x^2) f'(x^2)] + c \). This matches the expression given in option (C).

Key Concepts

Integration Techniques

Integration is a fundamental concept in calculus often used to find areas under curves, among other applications. In more advanced calculus, it's important to have a range of techniques to solve complex integrals efficiently. Here’s a look at some essential techniques:

• Substitution: This technique involves changing variables to simplify the integral. For example, if we have \( \int x f(x^2) \frac{1}{2} dt \), it becomes easier to evaluate by letting \( t = x^2 \).
• Partial Fractions: Useful when the integrand is a rational function, allowing you to express it as a sum of simpler fractions.
• Integration by Parts: Particularly useful when the integrand is a product of functions as shown in our problem.

By understanding and practicing these techniques, you can tackle a variety of integrals with more confidence.

Leibniz Rule

The Leibniz rule for differentiation under the integral sign is a powerful tool in calculus, particularly useful when dealing with integrals that involve a parameter. Although not directly used in this problem, understanding the concept enriches your calculus toolkit.
Leibniz's rule can be expressed as:\[\frac{d}{dt} \int_{a(t)}^{b(t)} f(x, t) \, dx = f(b(t), t)\cdot b'(t) - f(a(t), t)\cdot a'(t) + \int_{a(t)}^{b(t)} \frac{\partial}{\partial t} f(x, t) \, dx\]

• First Term: Shows the effect of the upper limit with respect to \( t \).
• Second Term: Shows the effect of the lower limit with respect to \( t \).
• Third Term: Integrates the partial derivative of \( f \) with respect to \( t \).

While Leibniz's rule is more suited to functions with variable limits or parameters, it's an excellent example of the elegance and power of derivative manipulations within integrals.

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation. It is particularly handy when tackling integrals of the form \( \int u \, dv \), where \( u \) and \( v \) are functions of \( x \). You can remember it easily through the formula:
\[\int u \, dv = uv - \int v \, du\]

• First Step: Identify parts of the integral that can be set as \( u \) and \( dv \). In our problem, recognizing parts that align with the derivative pattern helps in simplifying the expression.
• Next Step: Differentiate \( u \) to find \( du \), and integrate \( dv \) to get \( v \). Once you have these, apply the formula.
• Solve: Substituting back, you get a more manageable integral, possibly needing another application of integration by parts or other integration techniques.

In our problem, using integration by parts allows us to move towards the simpler form \( \frac{1}{2} [f(x^2) g'(x^2) - g(x^2) f'(x^2)] + c \). Hence, the application makes it more straightforward to compute specific integral forms, as displayed in our solution. Choosing the best parts to differentiate simplifies the integration process significantly.