JEE Main 11 Jan 2019 (Evening)

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Animated Solution for Mathematics - Indefinite Integration: 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:

Q: Why can't we just use substitution $u = 2x-1$?

You absolutely can! If you set $u = 2x-1$, then $du = 2dx$, so $dx = du/2$. Also, $x = (u+1)/2$. Substituting this into the integral $\int \frac{(u+1)/2 + 1}{\sqrt{u}} \cdot \frac{du}{2}$ will yield the same result. However, the algebraic manipulation method we used is often faster and helps you develop the intuition to 'see' the structure of the expression without writing out the full substitution.

Q: Why do we factor out $\sqrt{2x-1}$ at the end?

The question specifically asks for the form $f(x)\sqrt{2x-1} + C$. This is a structural constraint. We are not just solving the integral; we are solving for a specific representation of the answer. Factoring is the final step to align our result with the required format.

Select Answer:

Visualized Solution

Goal and Analysis

Goal: Find $f (x)$ if $\int \frac{x + 1}{2 x - 1} d x = f (x) 2 x - 1 + C$
Strategy: Express the numerator $(x + 1)$ in terms of the denominator's base $(2 x - 1)$ to simplify the integral.

Manipulating the Numerator: Step $1$

Multiply and divide the numerator by $2$ :
$\frac{x + 1}{2 x - 1} = \frac{\frac{1}{2} ( 2 x + 2 )}{2 x - 1}$

Manipulating the Numerator: Step $2$

Rewrite $2$ as $- 1 + 3$ :
$\frac{1}{2} (2 x + 2) = \frac{1}{2} (2 x - 1 + 3) = \frac{1}{2} (2 x - 1) + \frac{3}{2}$

Splitting the Integral

Substitute back and split:
$\int \frac{\frac{1}{2} ( 2 x - 1 ) + \frac{3}{2}}{2 x - 1} d x = \frac{1}{2} \int \frac{2 x - 1}{2 x - 1} d x + \frac{3}{2} \int \frac{1}{2 x - 1} d x$
$= \frac{1}{2} \int (2 x - 1)^{\frac{1}{2}} d x + \frac{3}{2} \int (2 x - 1)^{- \frac{1}{2}} d x$

Integrating the First Term

Apply Power Rule $\int (a x + b)^{n} d x = \frac{( a x + b ) ^{n + 1}}{a ( n + 1 )}$ to the first term:
$\frac{1}{2} \int (2 x - 1)^{\frac{1}{2}} d x = \frac{1}{2} \cdot \frac{( 2 x - 1 ) ^{\frac{3}{2}}}{2 \cdot \frac{3}{2}} = \frac{1}{6} (2 x - 1)^{\frac{3}{2}}$

Integrating the Second Term

Apply Power Rule to the second term:
$\frac{3}{2} \int (2 x - 1)^{- \frac{1}{2}} d x = \frac{3}{2} \cdot \frac{( 2 x - 1 ) ^{\frac{1}{2}}}{2 \cdot \frac{1}{2}} = \frac{3}{2} (2 x - 1)^{\frac{1}{2}}$

Combining the Results

Combine the integrated terms:
$\int \frac{x + 1}{2 x - 1} d x = \frac{1}{6} (2 x - 1)^{\frac{3}{2}} + \frac{3}{2} (2 x - 1)^{\frac{1}{2}} + C$

Factoring $2 x - 1$

Factor out $(2 x - 1)^{\frac{1}{2}}$ :
$= 2 x - 1 [\frac{1}{6} (2 x - 1) + \frac{3}{2}] + C$

Simplifying the Expression

Simplify the term inside the brackets:
$f (x) = \frac{2 x - 1}{6} + \frac{3}{2} = \frac{2 x - 1 + 9}{6} = \frac{2 x + 8}{6} = \frac{2 ( x + 4 )}{6} = \frac{x + 4}{3}$

Final Result: $f (x) = \frac{x + 4}{3}$

Final Answer: $f (x) = \frac{1}{3} (x + 4)$
This matches Option 3.

00:00 / 00:00

The Sigma Insight: Integration by Substitution

Analyzing the Setup

The integral we are tasked to solve is:

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

The denominator,

2 x - 1

, acts as our anchor. To simplify the integration, we must perform "algebraic surgery" on the numerator to make it reflect the structure of the denominator.

The Algebraic Transformation

We begin by manipulating the numerator

x + 1

to introduce the term

2 x - 1

. First, we multiply and divide by

2

x + 1 = \frac{1}{2} (2 x + 2)

Next, we rewrite the constant

2

- 1 + 3

to isolate the term

2 x - 1

x + 1 = \frac{1}{2} (2 x - 1 + 3) = \frac{1}{2} (2 x - 1) + \frac{3}{2}

Evaluating the Integrals

Substituting this back into the original integral, we split the expression into two manageable parts:

I = \frac{1}{2} \int 2 x - 1 d x + \frac{3}{2} \int (2 x - 1)^{- \frac{1}{2}} d x

Applying the power rule for integration, we obtain:

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

Simplifying the coefficients, we arrive at:

I = \frac{1}{6} (2 x - 1)^{\frac{3}{2}} + \frac{3}{2} (2 x - 1)^{\frac{1}{2}} + C

Final Simplification

To express the result in the form

f (x) 2 x - 1 + C

, we factor out

2 x - 1

I = 2 x - 1 [\frac{1}{6} (2 x - 1) + \frac{3}{2}] + C

Simplifying the expression inside the brackets:

\frac{2 x - 1 + 9}{6} = \frac{2 x + 8}{6} = \frac{x + 4}{3}

Thus, the final result is:

\frac{x + 4}{3} 2x - 1 + C

Common Traps & FAQs

Why can't we just use substitution

u = 2 x - 1

You absolutely can! If you set

u = 2 x - 1

, then

d u = 2 d x

, so

d x = d u /2

. Also,

x = (u + 1) /2

. Substituting this into the integral

\int \frac{( u + 1 ) /2 + 1}{u} \cdot \frac{d u}{2}

will yield the same result. However, the algebraic manipulation method we used is often faster and helps you develop the intuition to 'see' the structure of the expression without writing out the full substitution.

How did you know to rewrite

2

- 1 + 3

That is the 'Art of the Match.' We looked at the denominator

2 x - 1

and realized we needed a

2 x - 1

in the numerator. Since we had

2 x + 2

, we simply asked: 'What do I need to subtract from

2

to get

- 1

?' The answer is

3

. It is a deliberate, goal-oriented adjustment.

Why do we factor out

2 x - 1

at the end?

The question specifically asks for the form

f (x) 2 x - 1 + C

. This is a structural constraint. We are not just solving the integral; we are solving for a specific representation of the answer. Factoring is the final step to align our result with the required format.

Animated Solution for Mathematics - Indefinite Integration: 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:

Select Answer:

Visualized Solution

Goal and Analysis

Manipulating the Numerator: Step $1$

Manipulating the Numerator: Step $2$

Splitting the Integral

Integrating the First Term

Integrating the Second Term

Combining the Results

Factoring $2 x - 1$

Simplifying the Expression

Final Result: $f (x) = \frac{x + 4}{3}$

The Sigma Insight: Integration by Substitution

Analyzing the Setup

The Algebraic Transformation

Evaluating the Integrals

Final Simplification

Similar Questions

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Animated Solution for Mathematics - Indefinite Integration: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∫2x−1​x+1​dx=f(x)2x−1​+C, where C is a constant of integration, then f(x) is equal to:

Select Answer:

Visualized Solution

Goal and Analysis

Manipulating the Numerator: Step .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 1

Manipulating the Numerator: Step .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2

Splitting the Integral

Integrating the First Term

Integrating the Second Term

Combining the Results

Factoring .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2x−1​

Simplifying the Expression

Final Result: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(x)=3x+4​

The Sigma Insight: Integration by Substitution

Analyzing the Setup

The Algebraic Transformation

Evaluating the Integrals

Final Simplification

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Animated Solution for Mathematics - Indefinite Integration: 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:

Manipulating the Numerator: Step $1$

Manipulating the Numerator: Step $2$

Factoring $2 x - 1$

Final Result: $f (x) = \frac{x + 4}{3}$

If $f (\frac{x - 4}{x + 2}) = 2 x + 1, (x \in R - {1, - 2})$ , then $\int f (x) d x$ is equal to : (where C is a constant of integration)

The integral $\int \frac{1}{4 ( x - 1 ) ^{3} ( x + 2 ) ^{5}} d x$ is equal to : (where $C$ is a constant of integration)

If $\int \frac{d x}{x ^{3} ( 1 + x ^{6} ) ^{2/3}} = x f (x) (1 + x^{6})^{1/3} + C$ where C is a constant of integration, then the function f(x) is equal to-

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The integral $\int \frac{2 x ^{3} - 1}{x ^{4} + x} d x$ is equal to :

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