Problem 103 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\int f(x) d x=\Psi(x)\) then \(\int x^{5} f\left(x^{3}\right) d x\) is equal to (A) \(\frac{1}{3} x^{3} \Psi\left(x^{3}\right)-3 \int x^{3} \Psi\left(x^{3}\right) d x+C\) [2013] (B) \(\frac{1}{3} x^{3} \Psi\left(x^{3}\right)-\int x^{2} \Psi\left(x^{3}\right) d x+C\) (C) \(\frac{1}{3}\left[x^{3} \Psi\left(x^{3}\right)-\int x^{3} \Psi\left(x^{3}\right) d x\right]+C\) (D) \(\frac{1}{3}\left[x^{3} \Psi\left(x^{3}\right)-\int x^{2} \Psi\left(x^{3}\right) d x\right]+C\)

Step-by-Step Solution

Answer

Option D: \( \frac{1}{3} \left[ x^3 \Psi(x^3) - \int x^2 \Psi(x^3) dx \right] + C \) is correct.

Step 1: Identify the U-substitution

To solve the integral, begin by identifying the substitution that can simplify the expression. Notice that the integrand contains the function \( f(x^3) \). Set \( u = x^3 \), then \( du = 3x^2 dx \).

Step 2: Express the Integral in Terms of u

Rewrite the integral \( \int x^5 f(x^3) dx \) in terms of \( u \): With \( u = x^3 \) and \( x^5 = (x^3)^{5/3} = u^{5/3} \), the differential becomes \( dx = \frac{du}{3x^2} \). Thus, the integral becomes:\[ \int x^5 f(x^3) dx = \int x^3 x^2 f(x^3) dx = \frac{1}{3} \int u \cdot f(u) \cdot \frac{du}{x^2} \]

Step 3: Simplify Further

Substitute \( x^2 = u^{2/3} \) into the integral:\[ = \frac{1}{3} \int u f(u) \cdot \frac{du}{u^{2/3}} = \frac{1}{3} \int u^{1/3} f(u) du \]

Step 4: Integrate Using Integration by Parts

Apply integration by parts to \( \frac{1}{3} \int u^{1/3} f(u) du \), where\( v' = f(u) \rightarrow v = \Psi(u) \) (given that \( \int f(u) du = \Psi(u) \))and \( dv = u^{1/3} \rightarrow w = \frac{3}{4} u^{4/3} \). Integrating via parts formula: \( \int v \, dw = vw - \int w \, dv \), gives us:\[ \frac{1}{3} \left[ u^{1/3} \Psi(u) - \int \frac{3}{4} u^{4/3} \Psi(u) du \right] = \frac{1}{3} \left[ u \Psi(u) - \int \frac{3}{4} u \Psi(u) du \right] \]

Step 5: Substitute Back in Terms of x

Revert the substitution \( u = x^3 \):\[ \frac{1}{3} \left[ x^3 \Psi(x^3) - \int \frac{3}{4} x^3 \Psi(x^3) dx \right] = \frac{1}{3} \left[ x^3 \Psi(x^3) - \int x^2 \Psi(x^3) dx \right] \]

Step 6: Add the Constant of Integration

Finally, add the constant of integration \( C \):\[ \frac{1}{3} \left[ x^3 \Psi(x^3) - \int x^2 \Psi(x^3) dx \right] + C \]

Key Concepts

U-substitution in Integral Calculus

When faced with complex integrals, U-substitution can be a powerful tool to simplify the expression. In the context of our problem, the function inside the integral, \( f(x^3) \), suggests a potential substitution.
By choosing \( u = x^3 \), the integral becomes more manageable. But how does this work? The substitution changes not just the function, but also the differential \( dx \).
Let's break it down:

• Set \( u = x^3 \), which means \( du = 3x^2 \, dx \).
• This lets us express \( dx \) in terms of \( du \): \( dx = \frac{du}{3x^2} \).
• Thus, the original integral \( \int x^5 f(x^3) \, dx \) can be expressed as \( \frac{1}{3} \int u f(u) \frac{du}{x^2} \).

Using U-substitution transforms the problem into a form where other integration techniques, like integration by parts, can then be applied.

Integration by Parts

Integration by parts is another essential technique in integral calculus, often used when products of functions are involved. It relies on the integration formula: \( \int v \, dw = vw - \int w \, dv \).
In our case, after simplifying the integral using U-substitution, we end up with \( \frac{1}{3} \int u^{1/3} f(u) \, du \). Here, integration by parts comes into play.
Here's how it's done:

• Choose \( v' = f(u) \), making \( v = \Psi(u) \) because \( \int f(u) \, du = \Psi(u) \).
• Set \( dv = u^{1/3} \, du \), so \( w = \frac{3}{4} u^{4/3} \).
• Replace the integral using the integration by parts formula, resulting in:
  \( \frac{1}{3} \left[ u \Psi(u) - \int \frac{3}{4} u \Psi(u) \, du \right] \).

Integration by parts turns the problem into a subtraction, helping to integrate functions that multiply together.

Integral Transformation

Integral transformation is a process of converting an integral into a different form that is easier to evaluate. By exploiting transformations like U-substitution and integration by parts, we achieve this conversion.
When dealing with the original integral \( \int x^5 f(x^3) \, dx \):

• Start by transforming using U-substitution: Convert variables and expressions, transitioning to simpler terms.
• Follow through with integration by parts, another transformation, to dissect the integral further.
• Finally, substitute back into terms of the original variable, \( x \), to complete the transformation process.

For our problem, the final result after transformations and simplifications is:
\( \frac{1}{3} \left[ x^3 \Psi(x^3) - \int x^2 \Psi(x^3) \, dx \right] + C \).
This showcases how integral transformation can help navigate through complex integrals to arrive at a solution efficiently.