JEE Main 2020 (9 January Shift 1)

LEVELJEE Main

Animated Solution for Mathematics - Indefinite Integration: The integral $\int \frac{d x}{( x + 4 ) ^{8/7} ( x - 3 ) ^{6/7}}$ is equal to: (where C is a constant of integration)

Q: Why did we choose $t = \frac{x-3}{x+4}$ and not the other way around?

You could actually choose $t = \frac{x+4}{x-3}$, but it would just lead to a different power in the final result. The choice is guided by the structure of the integrand. We chose $\frac{x-3}{x+4}$ because it was already present as a fraction in our manipulated expression. Consistency is key!

Select Answer:

Visualized Solution

Analyze the Integrand

Given integral: $I = \int \frac{d x}{( x + 4 ) ^{8/7} ( x - 3 ) ^{6/7}}$
Observe the exponents: $\frac{8}{7} + \frac{6}{7} = \frac{14}{7} = 2$
This suggests we should aim for a substitution of the form $t = \frac{x - a}{x - b}$

Manipulate the Denominator

Multiply and divide the denominator by $(x + 4)^{6/7}$ :
$I = \int \frac{d x}{( x + 4 ) ^{8/7} \cdot ( x + 4 ) ^{6/7} \cdot \frac{( x - 3 ) ^{6/7}}{( x + 4 ) ^{6/7}}}$

Simplify Exponents

Combine the terms using exponent rules:
$(x + 4)^{8/7} \cdot (x + 4)^{6/7} = (x + 4)^{14/7} = (x + 4)^{2}$
The integral becomes: $I = \int \frac{d x}{( x + 4 ) ^{2} ( \frac{x - 3}{x + 4} ) ^{6/7}}$

Define Substitution $t$

Let $t = \frac{x - 3}{x + 4}$

Differentiate $t$ (Quotient Rule)

Differentiating $t$ with respect to $x$ using the quotient rule:
$\frac{d t}{d x} = \frac{( x + 4 ) ( 1 ) - ( x - 3 ) ( 1 )}{( x + 4 ) ^{2}}$

Simplify the Differential

Simplify the numerator:
$\frac{d t}{d x} = \frac{x + 4 - x + 3}{( x + 4 ) ^{2}} = \frac{7}{( x + 4 ) ^{2}}$
Rearranging for $d x$ : $\frac{d t}{7} = \frac{d x}{( x + 4 ) ^{2}}$

Substitute into the Integral

Substitute $t$ and $d t$ into the integral:
$I = \int \frac{1}{t ^{6/7}} \cdot \frac{d t}{7}$
$I = \frac{1}{7} \int t^{- 6/7} d t$

Integrate the Power Function

Apply the power rule $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$ :
$I = \frac{1}{7} [\frac{t ^{- 6/7 + 1}}{- 6/7 + 1}] + C$
$I = \frac{1}{7} [\frac{t ^{1/7}}{1/7}] + C$

Final Back-substitution

Cancel the constants and simplify:
$I = t^{1/7} + C$
Substitute $t = \frac{x - 3}{x + 4}$ back:
$I = (\frac{x - 3}{x + 4})^{1/7} + C$

Conclusion and Key Takeaway

Key Takeaway: For integrals of the form $\int \frac{d x}{( x - a ) ^{m} ( x - b ) ^{n}}$ where $m + n = 2$ , the substitution $t = \frac{x - a}{x - b}$ is usually effective.
Final Answer: $(\frac{x - 3}{x + 4})^{1/7} + C$
This matches Option 3.

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The Sigma Insight: Integration by Substitution

Analyzing the Setup

We are tasked with evaluating the integral:

I = \int \frac{d x}{( x + 4 ) ^{8/7} ( x - 3 ) ^{6/7}}

At first glance, the fractional powers

8/7

and

6/7

appear daunting. However, in JEE Advanced mathematics, such structures often hide a specific pattern.

Observe the sum of the exponents:

\frac{8}{7} + \frac{6}{7} = \frac{14}{7} = 2

Whenever an integral contains two linear factors in the denominator whose fractional exponents sum to

2

, it is a classic candidate for a linear fractional substitution.

The Algebraic Alchemy

To simplify the expression, we aim to group the terms

(x - 3)

and

(x + 4)

under a common exponent. We multiply and divide the denominator by

(x + 4)^{6/7}

I = \int \frac{d x}{( x + 4 ) ^{8/7} \cdot ( x + 4 ) ^{6/7} \cdot ( \frac{x - 3}{x + 4} ) ^{6/7}}

The product

(x + 4)^{8/7} \cdot (x + 4)^{6/7}

simplifies to

(x + 4)^{14/7}

, which is simply

(x + 4)^{2}

. The integral now takes the form:

I = \int \frac{d x}{( x + 4 ) ^{2} ( \frac{x - 3}{x + 4} ) ^{6/7}}

The Substitution

We define the substitution

t = \frac{x - 3}{x + 4}

. To find the differential

d t

, we apply the quotient rule:

\frac{d t}{d x} = \frac{( x + 4 ) ( 1 ) - ( x - 3 ) ( 1 )}{( x + 4 ) ^{2}} = \frac{x + 4 - x + 3}{( x + 4 ) ^{2}} = \frac{7}{( x + 4 ) ^{2}}

This yields the relation:

\frac{d t}{7} = \frac{d x}{( x + 4 ) ^{2}}

Substituting this into our integral, we eliminate

x

entirely in favor of

t

I = \int \frac{1}{t ^{6/7}} \cdot \frac{d t}{7}

The Final Victory

The integral is now reduced to a standard power rule application:

I = \frac{1}{7} \int t^{- 6/7} d t

Integrating

t^{- 6/7}

with respect to

t

gives:

I = \frac{1}{7} \cdot \frac{t ^{1/7}}{1/7} + C = t^{1/7} + C

Substituting back

t = \frac{x - 3}{x + 4}

, we arrive at the final result:

$I = (\frac{x - 3}{x + 4})^{1/7} + C$

This elegant result demonstrates that even the most intimidating exponents can be tamed by identifying the underlying algebraic structure.

Common Traps & FAQs

Wait, why do we add the exponents? Is that a rule?

It is not a universal rule, but it is a powerful heuristic for integrals of the form

\int \frac{d x}{( x - a ) ^{m} ( x - b ) ^{n}}

. When

m + n = 2

, it signals that the expression can be transformed into a rational function of a single variable

t = \frac{x - a}{x - b}

. It is a beautiful piece of algebraic symmetry that turns a complex radical expression into a simple power rule problem.

Why did we choose

t = \frac{x - 3}{x + 4}

and not the other way around?

You could actually choose

t = \frac{x + 4}{x - 3}

, but it would just lead to a different power in the final result. The choice is guided by the structure of the integrand. We chose

\frac{x - 3}{x + 4}

because it was already present as a fraction in our manipulated expression. Consistency is key!

What if the sum of the exponents was not 2? Would this method still work?

If the sum is not 2, this specific substitution will not simplify the integral into a basic power rule. You would likely need to explore other techniques like partial fractions, trigonometric substitution, or perhaps a different algebraic manipulation. Always check the sum of exponents first—it is the quickest way to decide your path.

Animated Solution for Mathematics - Indefinite Integration: The integral $\int \frac{d x}{( x + 4 ) ^{8/7} ( x - 3 ) ^{6/7}}$ is equal to: (where C is a constant of integration)

Select Answer:

Visualized Solution

Analyze the Integrand

Manipulate the Denominator

Simplify Exponents

Define Substitution $t$

Differentiate $t$ (Quotient Rule)

Simplify the Differential

Substitute into the Integral

Integrate the Power Function

Final Back-substitution

Conclusion and Key Takeaway

The Sigma Insight: Integration by Substitution

Analyzing the Setup

The Algebraic Alchemy

The Substitution

The Final Victory

Similar Questions

Find integration $\int \frac{d x}{( x - 3 ) ^{6/7} ( x + 4 ) ^{8/7}}$

The integral $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$ equals

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

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

The integral $\int \frac{( x ^{8} - x ^{2} ) d x}{( x ^{12} + 3 x ^{6} + 1 ) t a n ^{- 1} ( x ^{3} + \frac{1}{x ^{3}} )}$ equal to :

Find the indefinite integral $\int (\frac{1}{3 x + 4 x} + \frac{l n ( 1 + 6 x )}{3 x + x}) d x$

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

Evaluate the following $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$

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-

The integral $\int \frac{2 x ^{12} + 5 x ^{9}}{( x ^{5} + x ^{3} + 1 ) ^{3}} d x$ is equal to

Animated Solution for Mathematics - Indefinite Integration: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The integral ∫(x+4)8/7(x−3)6/7dx​ is equal to: (where C is a constant of integration)

Select Answer:

Visualized Solution

Analyze the Integrand

Manipulate the Denominator

Simplify Exponents

Define Substitution .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } t

Differentiate .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } t (Quotient Rule)

Simplify the Differential

Substitute into the Integral

Integrate the Power Function

Final Back-substitution

Conclusion and Key Takeaway

The Sigma Insight: Integration by Substitution

Analyzing the Setup

The Algebraic Alchemy

The Substitution

The Final Victory

Similar Questions

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Animated Solution for Mathematics - Indefinite Integration: The integral $\int \frac{d x}{( x + 4 ) ^{8/7} ( x - 3 ) ^{6/7}}$ is equal to: (where C is a constant of integration)

Define Substitution $t$

Differentiate $t$ (Quotient Rule)

Find integration $\int \frac{d x}{( x - 3 ) ^{6/7} ( x + 4 ) ^{8/7}}$

The integral $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$ equals

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

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

The integral $\int \frac{( x ^{8} - x ^{2} ) d x}{( x ^{12} + 3 x ^{6} + 1 ) t a n ^{- 1} ( x ^{3} + \frac{1}{x ^{3}} )}$ equal to :

Find the indefinite integral $\int (\frac{1}{3 x + 4 x} + \frac{l n ( 1 + 6 x )}{3 x + x}) d x$

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

Evaluate the following $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$

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-

The integral $\int \frac{2 x ^{12} + 5 x ^{9}}{( x ^{5} + x ^{3} + 1 ) ^{3}} d x$ is equal to