JEE Main 2019 (11 January)

LEVELJEE Main

Animated Solution for Mathematics - Indefinite Integration: If $\int \frac{1 - x ^{2}}{x ^{4}} d x = A (x) (1 - x^{2})^{m} + C$ , for a suitable chosen integer $m$ and a function $A (x)$ , where $C$ is a constant of integration then $(A (x))^{m}$ equals :

Q: How do I know what $m$ is? Is it always 3?

$m$ is not a constant; it is determined by the power of the radical in your final integrated expression. In this specific problem, the power rule for integration resulted in a $3/2$ power, which naturally leads to $m=3$. Do not guess $m$ beforehand; integrate first, simplify your result, and then match the form to find $m$.

Q: I got confused with the negative sign in the derivative. How do I keep track?

It is a common trap! When differentiating $x^{-2}$, the power rule gives $-2x^{-3}$. Always write out the full derivative step-by-step. If you lose a negative sign, your final answer will be off by a factor of $-1$, which is a classic distractor in multiple-choice questions. Treat the negative sign with the same respect as the variable itself.

Select Answer:

Progressive Hints

Hint 1

The Spark: Look at the denominator

x^{4}

and the radical

1 - x^{2}

. This is a classic JEE trap. You cannot integrate this directly. You must manipulate the expression inside the radical to create a term whose derivative is waiting for you in the denominator. Think about factoring out the highest power of

x

from inside the square root.

Hint 2

The Arsenal: Your primary weapon here is algebraic manipulation followed by a clever substitution. Factor out

x^{2}

from the term

1 - x^{2}

inside the square root. This will pull an

x

out of the radical, which will cancel with the

x^{4}

in the denominator, leaving you with a

1/ x^{3}

term. This is the 'golden ticket' because

1/ x^{3}

is the derivative of

1/ x^{2}

Hint 3

The Execution: Start your substitution by setting

t = \frac{1}{x ^{2}} - 1

. Differentiate this with respect to

x

to find

d t

. You will find that

\frac{1}{x ^{3}} d x = - \frac{1}{2} d t

. Substitute these into your integral to transform it into a simple power rule problem:

\int t \cdot (- \frac{1}{2}) d t

Visualized Solution

Understanding the Integral Form

Given Integral: $I = \int \frac{1 - x ^{2}}{x ^{4}} d x$
Target Form: $I = A (x) (1 - x^{2})^{m} + C$
Objective: Find $(A (x))^{m}$

Strategy for the Radical

Notice the term $1 - x^{2}$ and the denominator $x^{4}$ .
Standard technique: Factor out the highest power of $x$ from inside the square root.
This creates a term whose derivative is present outside.

Factoring out $x^{2}$

Inside the root: $1 - x^{2} = x^{2} (\frac{1}{x ^{2}} - 1)$
Apply the square root: $1 - x^{2} = x^{2} (\frac{1}{x ^{2}} - 1)$
Simplify: $1 - x^{2} = x \frac{1}{x ^{2}} - 1$

Simplifying the Integrand

Substitute back: $I = \int \frac{x \frac{1}{x ^{2}} - 1}{x ^{4}} d x$
Cancel $x$ from numerator and denominator.
Simplified Integral: $I = \int \frac{\frac{1}{x ^{2}} - 1}{x ^{3}} d x$

Choosing the Substitution

Let the term inside the root be $t$ .
Substitution: $t = \frac{1}{x ^{2}} - 1$
Why? Because the derivative of $x^{- 2}$ involves $x^{- 3}$ , which matches our denominator!

Differentiating the Substitution

Differentiate $t = x^{- 2} - 1$ with respect to $x$ .
$\frac{d t}{d x} = - 2 x^{- 3} = \frac{- 2}{x ^{3}}$
Rearrange to isolate the matching term: $\frac{1}{x ^{3}} d x = - \frac{1}{2} d t$

Transforming the Integral

Original: $I = \int \frac{1}{x ^{2}} - 1 \cdot \frac{1}{x ^{3}} d x$
Substitute $t$ and $d t$ : $I = \int t \cdot (- \frac{1}{2}) d t$
Pull out the constant: $I = - \frac{1}{2} \int t^{1/2} d t$

Executing the Integration

Apply power rule: $\int t^{n} d t = \frac{t ^{n + 1}}{n + 1}$
$I = - \frac{1}{2} [\frac{t ^{3/2}}{3/2}] + C$
Simplify the fraction: $I = - \frac{1}{2} \cdot \frac{2}{3} t^{3/2} + C$
Result: $I = - \frac{1}{3} t^{3/2} + C$

Back-Substitution

Recall our substitution: $t = \frac{1}{x ^{2}} - 1$
Substitute $t$ back into the result:
$I = - \frac{1}{3} (\frac{1}{x ^{2}} - 1)^{3/2} + C$
Take a common denominator: $I = - \frac{1}{3} (\frac{1 - x ^{2}}{x ^{2}})^{3/2} + C$

Matching the Target Form

Distribute the power $3/2$ to numerator and denominator.
Denominator: $(x^{2})^{3/2} = x^{3}$
Numerator: $(1 - x^{2})^{3/2} = (1 - x^{2})^{3}$
Rewrite $I$ : $I = - \frac{1}{3 x ^{3}} (1 - x^{2})^{3} + C$

Finding $(A (x))^{m}$

Compare with target: $I = A (x) (1 - x^{2})^{m} + C$
We find: $A (x) = - \frac{1}{3 x ^{3}}$ and $m = 3$
Calculate $(A (x))^{m} = (- \frac{1}{3 x ^{3}})^{3}$
Final Answer: $(A (x))^{m} = - \frac{1}{27 x ^{9}}$

00:00 / 00:00

The Sigma Insight: Integration by Substitution

Analyzing the Setup

We are tasked with evaluating the integral:

I = \int \frac{1 - x ^{2}}{x ^{4}} d x

At first glance, the combination of a square root and a high power of

x

in the denominator might suggest trigonometric substitution. However, for JEE Advanced efficiency, we look for a hidden algebraic structure.

Manipulating the Integrand

We begin by manipulating the expression inside the radical,

1 - x^{2}

. By factoring out

x^{2}

, we rewrite the term as

x^{2} (\frac{1}{x ^{2}} - 1)

Pulling

x^{2}

out of the square root yields

x

, transforming the integral into:

I = \int \frac{x \frac{1}{x ^{2}} - 1}{x ^{4}} d x = \int \frac{\frac{1}{x ^{2}} - 1}{x ^{3}} d x

The Substitution Strategy

This is the "Aha!" moment. We observe that the derivative of the inner function

u = \frac{1}{x ^{2}} - 1

involves

x^{- 3}

, which is exactly what we have in the denominator.

Let

t = \frac{1}{x ^{2}} - 1

. Then, the differential is:

d t = - 2 x^{- 3} d x \Rightarrow \frac{1}{x ^{3}} d x = - \frac{1}{2} d t

Substituting these into our integral, the expression collapses into a standard power rule form:

I = - \frac{1}{2} \int t d t

Final Calculation

Integrating with respect to

t

, we obtain:

I = - \frac{1}{2} \cdot \frac{t ^{3/2}}{3/2} + C = - \frac{1}{3} t^{3/2} + C

Substituting back

t = \frac{1 - x ^{2}}{x ^{2}}

, we get:

I = - \frac{1}{3} (\frac{1 - x ^{2}}{x ^{2}})^{3/2} + C = - \frac{1}{3 x ^{3}} (1 - x^{2})^{3} + C

Comparing this to the target form

A (x) (1 - x^{2})^{m} + C

, we identify $A (x) = - \frac{1}{3 x ^{3}}$ and $m = 3$ .

The final result of the requested operation

(A (x))^{m}

is:

(A (x))^{m} = (- \frac{1}{3 x ^{3}})^{3} = - \frac{1}{27 x ^{9}}

Common Traps & FAQs

Why did we factor out

x^{2}

from the root? Why not just use trigonometric substitution like

x = sin θ

Trigonometric substitution is a valid path, but it often leads to complex trigonometric integrals that take longer to solve. Factoring out

x^{2}

is an algebraic shortcut that directly reveals the derivative of the inner function, making the integration much faster. In JEE Advanced, time is your greatest asset, so always look for the most direct algebraic path first.

How do I know what

m

is? Is it always 3?

m

is not a constant; it is determined by the power of the radical in your final integrated expression. In this specific problem, the power rule for integration resulted in a

3/2

power, which naturally leads to

m = 3

. Do not guess

m

beforehand; integrate first, simplify your result, and then match the form to find

m

I got confused with the negative sign in the derivative. How do I keep track?

It is a common trap! When differentiating

x^{- 2}

, the power rule gives

- 2 x^{- 3}

. Always write out the full derivative step-by-step. If you lose a negative sign, your final answer will be off by a factor of

- 1

, which is a classic distractor in multiple-choice questions. Treat the negative sign with the same respect as the variable itself.

Animated Solution for Mathematics - Indefinite Integration: If $\int \frac{1 - x ^{2}}{x ^{4}} d x = A (x) (1 - x^{2})^{m} + C$ , for a suitable chosen integer $m$ and a function $A (x)$ , where $C$ is a constant of integration then $(A (x))^{m}$ equals :

Select Answer:

Visualized Solution

Understanding the Integral Form

Strategy for the Radical

Factoring out $x^{2}$

Simplifying the Integrand

Choosing the Substitution

Differentiating the Substitution

Transforming the Integral

Executing the Integration

Back-Substitution

Matching the Target Form

Finding $(A (x))^{m}$

The Sigma Insight: Integration by Substitution

Analyzing the Setup

Manipulating the Integrand

The Substitution Strategy

Final Calculation

Similar Questions

$\int \frac{x ^{2} - 1}{x ^{3} 2 x ^{4} - 2 x ^{2} + 1} d x =$

If $\int \frac{x + 1}{2 x - 1} d x = f (x) 2 x - 1 + C$ , where $C$ is a constant of integration, then $f (x)$ is equal to:

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Select Answer:

Visualized Solution

Understanding the Integral Form

Strategy for the Radical

Factoring out .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x2

Simplifying the Integrand

Choosing the Substitution

Differentiating the Substitution

Transforming the Integral

Executing the Integration

Back-Substitution

Matching the Target Form

Finding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (A(x))m

The Sigma Insight: Integration by Substitution

Analyzing the Setup

Manipulating the Integrand

The Substitution Strategy

Final Calculation

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Finding $(A (x))^{m}$

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