JEE Main 2023 (24 January Shift 1)

LEVELJEE Main

Animated Solution for Mathematics - Definite Integration: The value of $\frac{8}{π} \int_{0}^{π /2} \frac{( c o s x ) ^{2023}}{( s i n x ) ^{2023} + ( c o s x ) ^{2023}} d x$ is ______.

Q: Why does the power 2023 not matter in the final answer?

The power 2023 is a distraction! Because the denominator $(\sin x)^{2023} + (\cos x)^{2023}$ is symmetric with respect to the substitution $x \to \frac{\pi}{2} - x$, the numerator simply swaps from $(\cos x)^{2023}$ to $(\sin x)^{2023}$. When you add the two integrals, the numerator becomes the same as the denominator, which cancels out to 1. This happens for any power $n$!

Q: Can I use this property for any limits?

No, the King's Property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$ is specifically designed for integrals starting from zero to some value $a$. If the lower limit is not zero, you cannot use this specific form of the property directly.

Enter Numerical Value:

Visualized Solution

Defining the Integral $I$

Let $I = \frac{8}{π} \int_{0}^{π /2} \frac{( c o s x ) ^{2023}}{( s i n x ) ^{2023} + ( c o s x ) ^{2023}} d x$
This is our starting point, labeled as Equation (1).

The King's Property

Apply the property: $\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$

Applying the Substitution

Here, $a = \frac{π}{2}$ , so we replace $x$ with $(\frac{π}{2} - x)$ .
$I = \frac{8}{π} \int_{0}^{π /2} \frac{( c o s ( \frac{π}{2} - x ) ) ^{2023}}{( s i n ( \frac{π}{2} - x ) ) ^{2023} + ( c o s ( \frac{π}{2} - x ) ) ^{2023}} d x$

Trigonometric Transformation

Using identities: $cos (\frac{π}{2} - x) = sin x$ and $sin (\frac{π}{2} - x) = cos x$
$I = \frac{8}{π} \int_{0}^{π /2} \frac{( s i n x ) ^{2023}}{( c o s x ) ^{2023} + s i n x ) ^{2023}} d x$
This is Equation (2).

Adding the Two Integrals

Add Equation (1) and Equation (2):
$2 I = \frac{8}{π} \int_{0}^{π /2} (\frac{( c o s x ) ^{2023}}{( s i n x ) ^{2023} + ( c o s x ) ^{2023}} + \frac{( s i n x ) ^{2023}}{( s i n x ) ^{2023} + ( c o s x ) ^{2023}}) d x$

Simplifying the Integrand

Combine the fractions:
$2 I = \frac{8}{π} \int_{0}^{π /2} \frac{( c o s x ) ^{2023} + ( s i n x ) ^{2023}}{( s i n x ) ^{2023} + ( c o s x ) ^{2023}} d x$
$2 I = \frac{8}{π} \int_{0}^{π /2} 1 d x$

Evaluating the Integral

Integrate: $2 I = \frac{8}{π} [x]_{0}^{π /2}$
Substitute limits: $2 I = \frac{8}{π} (\frac{π}{2} - 0)$

Final Calculation for $2 I$

$2 I = \frac{8}{π} \cdot \frac{π}{2}$
$2 I = 4$

Solving for $I$

$I = \frac{4}{2}$
Final Answer: $I = 2$

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The Sigma Insight: Fundamental Theorem & Properties of Definite Integrals

Solution Diagram

Analyzing the Setup

We are tasked with evaluating the definite integral:

I = \frac{8}{π} \int_{0}^{π /2} \frac{( cos x ) ^{2023}}{( sin x ) ^{2023} + ( cos x ) ^{2023}} d x

The exponent of

2023

appears daunting, but we can simplify this using the King's Property of definite integrals.

Applying the King's Property

The King's Property states that:

\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x

By applying this property with

a = \frac{π}{2}

and replacing

x

with

\frac{π}{2} - x

, we utilize the trigonometric identities

cos (\frac{π}{2} - x) = sin x

and

sin (\frac{π}{2} - x) = cos x

This transformation yields a new expression for the same integral:

I = \frac{8}{π} \int_{0}^{π /2} \frac{( sin x ) ^{2023}}{( cos x ) ^{2023} + ( sin x ) ^{2023}} d x

The Master Equation

Now, we add the original integral

I

to the transformed integral

I

2 I = \frac{8}{π} \int_{0}^{π /2} \frac{( cos x ) ^{2023} + ( sin x ) ^{2023}}{( sin x ) ^{2023} + ( cos x ) ^{2023}} d x

Since the numerator and denominator are identical, the integrand simplifies to

1

2 I = \frac{8}{π} \int_{0}^{π /2} 1 d x

Final Calculation

Evaluating the integral of

1

from

0

\frac{π}{2}

gives us

\frac{π}{2}

2 I = \frac{8}{π} \cdot \frac{π}{2}

2 I = 4

Dividing by

2

, we arrive at the final result:

$I = 2$

Common Traps & FAQs

Why does the power 2023 not matter in the final answer?

The power 2023 is a distraction! Because the denominator

(sin x)^{2023} + (cos x)^{2023}

is symmetric with respect to the substitution

x \to \frac{π}{2} - x

, the numerator simply swaps from

(cos x)^{2023}

(sin x)^{2023}

. When you add the two integrals, the numerator becomes the same as the denominator, which cancels out to 1. This happens for any power

n

Can I use this property for any limits?

No, the King's Property

\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x

is specifically designed for integrals starting from zero to some value

a

. If the lower limit is not zero, you cannot use this specific form of the property directly.

Why did we add the two integrals

I + I

We add them because the resulting integrands have the same denominator. Adding them allows us to combine the numerators, which in this case perfectly match the denominator, simplifying the entire expression to 1. It is a standard technique to solve integrals that are otherwise impossible to evaluate directly.

Animated Solution for Mathematics - Definite Integration: The value of $\frac{8}{π} \int_{0}^{π /2} \frac{( c o s x ) ^{2023}}{( s i n x ) ^{2023} + ( c o s x ) ^{2023}} d x$ is ______.

Enter Numerical Value:

Visualized Solution

Defining the Integral $I$

The King's Property

Applying the Substitution

Trigonometric Transformation

Adding the Two Integrals

Simplifying the Integrand

Evaluating the Integral

Final Calculation for $2 I$

Solving for $I$

The Sigma Insight: Fundamental Theorem & Properties of Definite Integrals

Analyzing the Setup

Applying the King's Property

The Master Equation

Final Calculation

Similar Questions

The value of $\int_{0}^{2 π} \frac{x s i n ^{8} x}{s i n ^{8} x + c o s ^{8} x} d x$ is equal to:

The value of the integral $\frac{48}{π ^{4}} \int_{0}^{π} x^{2} (\frac{3 π x ^{2}}{2} - x^{3}) \frac{s i n x}{1 + c o s ^{2} x} d x$ is equal to ______.

The value of $\int_{- π /2}^{π /2} (\frac{1 + s i n ^{2} x}{1 + π ^{s i n x}}) d x$ is

$\frac{120}{π ^{3}} \int_{0}^{π} \frac{x ^{2} s i n x c o s x}{s i n ^{4} x + c o s ^{4} x} d x$ is equal to _______.

The value of the integral $\int_{- π /4}^{π /4} \frac{x + \frac{π}{4}}{2 - c o s 2 x} d x$ is

The integral $\frac{24}{π} \int_{0}^{2} \frac{( 2 - x ^{2} ) d x}{( 2 + x ^{2} ) 4 + x ^{4}}$ is equal to ______.

Determine the value of $\int_{- π}^{π} \frac{2 x ( 1 + s i n x )}{1 + c o s ^{2} x} d x$ .

The integral $\int_{0}^{π} \frac{8 x d x}{4 c o s ^{2} x + s i n ^{2} x}$ is equal to

The value of $\int_{- π}^{π} \frac{2 y ( 1 + s i n y )}{1 + c o s ^{2} y} d y$ is :

The value of the integral $\int_{0}^{π /2} \frac{3 c o s θ}{( c o s θ + s i n θ ) ^{5}} d θ$ equals

Animated Solution for Mathematics - Definite Integration: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of π8​∫0π/2​(sinx)2023+(cosx)2023(cosx)2023​dx is ______.

Enter Numerical Value:

Visualized Solution

Defining the Integral .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } I

The King's Property

Applying the Substitution

Trigonometric Transformation

Adding the Two Integrals

Simplifying the Integrand

Evaluating the Integral

Final Calculation for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 2I

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } I

The Sigma Insight: Fundamental Theorem & Properties of Definite Integrals

Analyzing the Setup

Applying the King's Property

The Master Equation

Final Calculation

Similar Questions

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Animated Solution for Mathematics - Definite Integration: The value of $\frac{8}{π} \int_{0}^{π /2} \frac{( c o s x ) ^{2023}}{( s i n x ) ^{2023} + ( c o s x ) ^{2023}} d x$ is ______.

Defining the Integral $I$

Final Calculation for $2 I$

Solving for $I$

The value of $\int_{0}^{2 π} \frac{x s i n ^{8} x}{s i n ^{8} x + c o s ^{8} x} d x$ is equal to:

The value of the integral $\frac{48}{π ^{4}} \int_{0}^{π} x^{2} (\frac{3 π x ^{2}}{2} - x^{3}) \frac{s i n x}{1 + c o s ^{2} x} d x$ is equal to ______.

The value of $\int_{- π /2}^{π /2} (\frac{1 + s i n ^{2} x}{1 + π ^{s i n x}}) d x$ is

$\frac{120}{π ^{3}} \int_{0}^{π} \frac{x ^{2} s i n x c o s x}{s i n ^{4} x + c o s ^{4} x} d x$ is equal to _______.

The value of the integral $\int_{- π /4}^{π /4} \frac{x + \frac{π}{4}}{2 - c o s 2 x} d x$ is

The integral $\frac{24}{π} \int_{0}^{2} \frac{( 2 - x ^{2} ) d x}{( 2 + x ^{2} ) 4 + x ^{4}}$ is equal to ______.

Determine the value of $\int_{- π}^{π} \frac{2 x ( 1 + s i n x )}{1 + c o s ^{2} x} d x$ .

The integral $\int_{0}^{π} \frac{8 x d x}{4 c o s ^{2} x + s i n ^{2} x}$ is equal to

The value of $\int_{- π}^{π} \frac{2 y ( 1 + s i n y )}{1 + c o s ^{2} y} d y$ is :

The value of the integral $\int_{0}^{π /2} \frac{3 c o s θ}{( c o s θ + s i n θ ) ^{5}} d θ$ equals