JEE Main 2019 (9 January)

LEVELJEE Main

Animated Solution for Mathematics - Indefinite Integration: If $f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 1 + 2 x ^{7} ) ^{2}} d x, (x \geq 0)$ and $f (0) = 0$ , then the value of $f (1)$ is :

Q: Wait, why did we divide by $x^{14}$? Why not just $x^7$?

That is a brilliant question! Remember, the denominator is $(x^2 + 1 + 2x^7)^2$. If we only divided by $x^7$, we would still have an $x^7$ inside the square, which doesn't help us simplify the expression outside. By dividing by $x^{14}$, we are essentially dividing by $(x^7)^2$. This allows us to bring the $x^7$ *inside* the square, effectively dividing each term inside the bracket by $x^7$. It is all about algebraic balance!

Q: Why do we need to find the constant $C$ using $f(0)=0$? Can't we just ignore it?

In indefinite integration, $C$ is just a constant. But here, we have a specific condition: $f(0) = 0$. This turns our indefinite integral into a specific function. If you ignore $C$, you are essentially saying the function passes through the origin, which might not be true for the general family of curves. Always use the boundary conditions provided to pin down the exact curve!

Select Answer:

Progressive Hints

Hint 1

Hint 1: Look at the denominator. It is a polynomial raised to a power. In JEE problems of this type, the goal is to transform the expression so that the numerator becomes the derivative of the 'inner' part of the denominator. Don't be intimidated by the powers; look for the hidden relationship between the degrees of the numerator and the denominator.

Hint 2

Hint 2: The key is to manipulate the denominator's bracket. Identify the highest power of

x

inside the bracket, which is

x^{7}

. When you factor this out, remember that the entire bracket is squared, so you are effectively pulling out

(x^{7})^{2} = x^{14}

. This is your golden key.

Hint 3

Hint 3: Divide both the numerator and the denominator by

x^{14}

. This will transform the integral into a form where you can set

u = x^{- 5} + x^{- 7} + 2

. Once you differentiate this

u

, you will see the numerator appear almost magically as

- d u

Visualized Solution

Analyze the Integral Structure

Given integral: $f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 1 + 2 x ^{7} ) ^{2}} d x$
Condition: $f (0) = 0$
Goal: Find the value of $f (1)$

The Algebraic Trick

Identify the highest power of $x$ inside the denominator's bracket: $x^{7}$
When taken out of the square, it becomes $(x^{7})^{2} = x^{14}$
Strategy: Divide numerator and denominator by $x^{14}$

Divide by $x^{14}$

$f (x) = \int \frac{\frac{5 x ^{8} + 7 x ^{6}}{x ^{14}}}{\frac{( x ^{2} + 1 + 2 x ^{7} ) ^{2}}{x ^{14}}} d x$

Simplify the Numerator

$\frac{5 x ^{8}}{x ^{14}} + \frac{7 x ^{6}}{x ^{14}}$
$= 5 x^{- 6} + 7 x^{- 8}$

Simplify the Denominator

$\frac{( x ^{2} + 1 + 2 x ^{7} ) ^{2}}{( x ^{7} ) ^{2}}$
$= (\frac{x ^{2}}{x ^{7}} + \frac{1}{x ^{7}} + \frac{2 x ^{7}}{x ^{7}})^{2}$
$= (x^{- 5} + x^{- 7} + 2)^{2}$

The Substitution

Let $u = x^{- 5} + x^{- 7} + 2$

Differentiate the Substitution

Differentiating both sides:
$d u = (- 5 x^{- 6} - 7 x^{- 8}) d x$
$- (5 x^{- 6} + 7 x^{- 8}) d x = d u$
$(5 x^{- 6} + 7 x^{- 8}) d x = - d u$

Rewrite the Integral

Substitute $u$ and $d u$ into the integral:
$f (x) = \int \frac{- d u}{u ^{2}}$
$f (x) = - \int u^{- 2} d u$

Perform the Integration

Integrating using power rule: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1}$
$f (x) = - (\frac{u ^{- 2 + 1}}{- 2 + 1}) + C$
$f (x) = - (\frac{u ^{- 1}}{- 1}) + C$
$f (x) = \frac{1}{u} + C$

Substitute Back $x$

Substitute back $u = x^{- 5} + x^{- 7} + 2$ :
$f (x) = \frac{1}{x ^{- 5} + x ^{- 7} + 2} + C$

Simplify $f (x)$

$f (x) = \frac{1}{\frac{1}{x ^{5}} + \frac{1}{x ^{7}} + 2} + C$
Multiply numerator and denominator by $x^{7}$ :
$f (x) = \frac{x ^{7}}{x ^{2} + 1 + 2 x ^{7}} + C$

Find the Constant $C$

Use the given condition: $f (0) = 0$
$0 = \frac{0 ^{7}}{0 ^{2} + 1 + 2 ( 0 ) ^{7}} + C$
$0 = \frac{0}{1} + C$
$C = 0$
So, $f (x) = \frac{x ^{7}}{x ^{2} + 1 + 2 x ^{7}}$

Calculate $f (1)$

Goal: Find $f (1)$
$f (1) = \frac{1 ^{7}}{1 ^{2} + 1 + 2 ( 1 ) ^{7}}$
$f (1) = \frac{1}{1 + 1 + 2}$
$f (1) = \frac{1}{4}$

Final Answer & Takeaway

Final Answer: $\frac{1}{4}$
Key Takeaway: For integrals of the form $\int \frac{a x ^{m} + b x ^{n}}{( c x ^{p} + d x ^{q} + ... ) ^{k}} d x$ , try factoring out the highest power of $x$ from the denominator bracket to create a substitution in the numerator.

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The Sigma Insight: Integration by Substitution

Analyzing the Setup

We are tasked with evaluating the integral:

f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 1 + 2 x ^{7} ) ^{2}} d x

At first glance, the expression appears chaotic due to the high powers and the squared polynomial in the denominator. However, in JEE Advanced mathematics, such complexity is often a mask for underlying elegance.

The Algebraic Insight

When a polynomial is raised to a power in the denominator, we should look for the derivative of the "inner" function. We examine the term inside the bracket:

(x^{2} + 1 + 2 x^{7})

The highest power present is

x^{7}

. Since the entire bracket is squared, factoring out

x^{7}

effectively pulls out

(x^{7})^{2} = x^{14}

. We proceed by dividing both the numerator and the denominator by

x^{14}

The numerator transforms as follows:

\frac{5 x ^{8}}{x ^{14}} + \frac{7 x ^{6}}{x ^{14}} = 5 x^{- 6} + 7 x^{- 8}

The Transformation

Next, we manipulate the denominator by bringing the division inside the square:

\frac{( x ^{2} + 1 + 2 x ^{7} ) ^{2}}{( x ^{7} ) ^{2}} = (\frac{x ^{2}}{x ^{7}} + \frac{1}{x ^{7}} + \frac{2 x ^{7}}{x ^{7}})^{2} = (x^{- 5} + x^{- 7} + 2)^{2}

We now define the substitution

u = x^{- 5} + x^{- 7} + 2

. Differentiating with respect to

x

yields:

d u = (- 5 x^{- 6} - 7 x^{- 8}) d x

Factoring out the negative sign, we obtain

- (5 x^{- 6} + 7 x^{- 8}) d x = d u

. This matches our numerator perfectly, allowing the integral to collapse into:

\int - u^{- 2} d u

The Integration and Final Polish

The integration is now straightforward. The integral of

- u^{- 2}

u^{- 1} + C

, which is equivalent to:

f (x) = \frac{1}{u} + C = \frac{1}{x ^{- 5} + x ^{- 7} + 2} + C

To simplify, we multiply the numerator and denominator by

x^{7}

f (x) = \frac{x ^{7}}{1 + x ^{2} + 2 x ^{7}} + C

Given the condition

f (0) = 0

, we find that

C = 0

. Finally, evaluating at

x = 1

f (1) = \frac{1 ^{7}}{1 ^{2} + 1 + 2 ( 1 ) ^{7}} = \frac{1}{4}

The final result is $\frac{1}{4}$ .

Common Traps & FAQs

Wait, why did we divide by

x^{14}

? Why not just

x^{7}

That is a brilliant question! Remember, the denominator is

(x^{2} + 1 + 2 x^{7})^{2}

. If we only divided by

x^{7}

, we would still have an

x^{7}

inside the square, which doesn't help us simplify the expression outside. By dividing by

x^{14}

, we are essentially dividing by

(x^{7})^{2}

. This allows us to bring the

x^{7}

*inside* the square, effectively dividing each term inside the bracket by

x^{7}

. It is all about algebraic balance!

What if the derivative of the denominator doesn't match the numerator exactly? Do I just give up?

Never give up! In JEE Advanced, these problems are designed with precision. If the derivative doesn't match perfectly, check your algebra. Often, you might have missed a constant factor or a sign. If it *still* doesn't match, you might need to split the integral into two parts or use a different substitution. But 99% of the time, the 'highest power' trick is the intended path.

Why do we need to find the constant

C

using

f (0) = 0

? Can't we just ignore it?

In indefinite integration,

C

is just a constant. But here, we have a specific condition:

f (0) = 0

. This turns our indefinite integral into a specific function. If you ignore

C

, you are essentially saying the function passes through the origin, which might not be true for the general family of curves. Always use the boundary conditions provided to pin down the exact curve!

Animated Solution for Mathematics - Indefinite Integration: If $f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 1 + 2 x ^{7} ) ^{2}} d x, (x \geq 0)$ and $f (0) = 0$ , then the value of $f (1)$ is :

Select Answer:

Visualized Solution

Analyze the Integral Structure

The Algebraic Trick

Divide by $x^{14}$

Simplify the Numerator

Simplify the Denominator

The Substitution

Differentiate the Substitution

Rewrite the Integral

Perform the Integration

Substitute Back $x$

Simplify $f (x)$

Find the Constant $C$

Calculate $f (1)$

Final Answer & Takeaway

The Sigma Insight: Integration by Substitution

Analyzing the Setup

The Algebraic Insight

The Transformation

The Integration and Final Polish

Similar Questions

If $f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 1 + 2 x ^{7} ) ^{2}} d x, (x \geq 0), f (0) = 0$ and $f (1) = \frac{1}{K}$ , then the value of $K$ is

If $f (x) = \int \frac{1}{x ^{1/4} ( 1 + x ^{1/4} )} d x$ , $f (0) = - 6,$ then $f (1)$ is equal to:

Let $f (x) = \int x^{3} 3 - x^{2} d x$ . If $5 f (2) = - 4$ , then $f (1)$ is equal to

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-

Let $f (x) = \int \frac{x}{( 1 + x ) ^{2}} d x (x > 0)$ . Then $f (3) - f (1)$ is equal to :

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

If $\int \frac{1}{x} \frac{1 - x}{1 + x} d x = g (x) + c, g (1) = 0$ , then $g (\frac{1}{2})$ is equal to :

Let $I (x) = \int \frac{x + 1}{x ( 1 + x e ^{x} ) ^{2}} d x, x > 0$ . If $lim_{x \to \infty} I (x) = 0$ then $I (1)$ is equal to

Evaluate $\int \frac{( x + 1 )}{x ( 1 + x e ^{x} ) ^{2}} d x$

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

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 f(x)=∫(x2+1+2x7)25x8+7x6​dx,(x≥0) and f(0)=0, then the value of f(1) is :

Select Answer:

Visualized Solution

Analyze the Integral Structure

The Algebraic Trick

Divide by .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x14

Simplify the Numerator

Simplify the Denominator

The Substitution

Differentiate the Substitution

Rewrite the Integral

Perform the Integration

Substitute Back .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x

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

Find the Constant .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C

Calculate .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(1)

Final Answer & Takeaway

The Sigma Insight: Integration by Substitution

Analyzing the Setup

The Algebraic Insight

The Transformation

The Integration and Final Polish

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)=∫(x2+1+2x7)25x8+7x6​dx,(x≥0),f(0)=0 and f(1)=K1​, then the value of K is

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)=∫(1+x)2x​​dx(x>0). Then f(3)−f(1) is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate the following ∫x2(x4+1)3/4dx​

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The integral ∫x2(x4+1)3/4dx​ equals

Animated Solution for Mathematics - Indefinite Integration: If $f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 1 + 2 x ^{7} ) ^{2}} d x, (x \geq 0)$ and $f (0) = 0$ , then the value of $f (1)$ is :

Divide by $x^{14}$

Substitute Back $x$

Simplify $f (x)$

Find the Constant $C$

Calculate $f (1)$

If $f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 1 + 2 x ^{7} ) ^{2}} d x, (x \geq 0), f (0) = 0$ and $f (1) = \frac{1}{K}$ , then the value of $K$ is

If $f (x) = \int \frac{1}{x ^{1/4} ( 1 + x ^{1/4} )} d x$ , $f (0) = - 6,$ then $f (1)$ is equal to:

Let $f (x) = \int x^{3} 3 - x^{2} d x$ . If $5 f (2) = - 4$ , then $f (1)$ is equal to

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-

Let $f (x) = \int \frac{x}{( 1 + x ) ^{2}} d x (x > 0)$ . Then $f (3) - f (1)$ is equal to :

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

If $\int \frac{1}{x} \frac{1 - x}{1 + x} d x = g (x) + c, g (1) = 0$ , then $g (\frac{1}{2})$ is equal to :

Let $I (x) = \int \frac{x + 1}{x ( 1 + x e ^{x} ) ^{2}} d x, x > 0$ . If $lim_{x \to \infty} I (x) = 0$ then $I (1)$ is equal to

Evaluate $\int \frac{( x + 1 )}{x ( 1 + x e ^{x} ) ^{2}} d x$

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