If ∫( x)(-11)/(2)( x)(-5)/(2)dx= - p₁q₁( x)(9)/(2)- p₂q₂( x)(5)/(2)- p₃q₃( x)(1)/(2)+ p₄q₄( x)(-3)/(2)+C$\int(\sin x)^{\frac{-11}{2}}(\cos x)^{\frac{-5}{2}}dx= -\frac{p_{1}}{q_{1}}(\cot x)^{\frac{9}{2}}-\frac{p_{2}}{q_{2}}(\cot x)^{\frac{5}{2}}-\frac{p_{3}}{q_{3}}(\cot x)^{\frac{1}{2}}+\frac{p_{4}}{q_{4}}(\cot x)^{\frac{-3}{2}}+C$, where pᵢ$p_{i}$ and qᵢ$q_{i}$ are positive integers with (pᵢ, qᵢ)=1$\gcd(p_{i}, q_{i})=1$ for i=1, 2, 3, 4 and C is the constant of integration, then 15p₁p₂p₃p₄q₁q₂q₃q₄$\frac{15p_{1}p_{2}p_{3}p_{4}}{q_{1}q_{2}q_{3}q_{4}}$ is equal to ____.

Related Formula

Area enclosed between two symmetric functions across four quadrants:

Core Logic

The curves are symmetric across the axes. The region is enclosed between the unit circle x² + y² = 1$x^2 + y^2 = 1$ and the parabolas y = ±(1 - x²)$y = \pm(1 - x^2)$.

By symmetry, the area α$\alpha$ in the first quadrant is the area under the circle minus the area under the parabola from x=0$x=0$ to x=1$x=1$.

Step 1: Calculate the Area Components

Area of a circle quadrant with r=1$r=1$ is (π)/(4)$\frac{\pi}{4}$.

Area under the parabola:

Step 2: Evaluate the Total Area

Multiplying both sides by 9:

Comparing with 9α = βπ + γ$9\alpha = \beta\pi + \gamma$, we get:

Hence,

Pattern Recognition

Symmetry helps divide the integration work by 4. Recognizing standard shapes (like a circle sector) allows you to bypass complex integration entirely for half of the problem.

Chapter Mix

Class 12 Mathematics: Integral Calculus

Related Formula

Additivity property of definite integrals over split interval boundary points:

Core Logic

Separate the integration tracking flow into two component operations:

Step 1: Solve Modulus Integral Term

The argument changes sign inside absolute value bounds when 4x - (pi)/(12) = 0 x = (pi)/(48)$4x - \frac{pi}{12} = 0 \implies x = \frac{pi}{48}$:

Evaluating across numerical boundaries gives:

Step 2: Solve Greatest Integer Component and Combine

Analyze step limits inside greatest integer block [2 x]$[2\sin x]$: For x in [0, (pi)/(6))$x \in \left[0, \frac{pi}{6}\right)$, 0 ≤ 2 x < 1 [2 x] = 0$0 \le 2\sin x < 1 \implies [2\sin x] = 0$. For x in [(pi)/(6), (pi)/(4)]$x \in \left[\frac{pi}{6}, \frac{pi}{4}\right]$, 1 ≤ 2 x < √(2) [2 x] = 1$1 \le 2\sin x < \sqrt{2} \implies [2\sin x] = 1$.

Combine both sections aggregated by multiplier 24:

Comparing directly with expression statement parameters 2π + α$2\pi + \alpha$ isolates response value: α = 12$\alpha = 12$

Pattern Recognition

Modulus arguments and step functions require isolating inflection transition numbers directly to break integrations up neatly.

Chapter Mix

Class 12 Mathematics: Definite Integration

Related Formula

King's property for definite integration:

Core Logic

Apply the identity x → -x$x \to -x$ to the integral:

Step 1: Adding both integral variations

Adding the equations eliminates the exponential denominator term (1+e^x$1+e^x$):

Step 2: Evaluating the integrated components

Integrating by parts gives:

Matching coefficients with the template: α = 2$\alpha = 2$ and \beta = -12.

Pattern Recognition

Exponential denominators like 1+e^x$1+e^x$ in symmetric integral intervals are prime candidates for simplification using King's property.

Chapter Mix

Class 12 Maths: Definite Integrals

Related Formula

Method of algebraic parameter substitution: Set 3-x² = t² -2xdx = 2tdt xdx = -tdt$3-x^2 = t^2 \implies -2x\mathrm{d}x = 2t\mathrm{d}t \implies x\mathrm{d}x = -t\mathrm{d}t$

Core Logic

Perform the specified variable parameter replacement steps:

Rewrite the internal integral block components:

Return to original reference variable x$x$:

Step 1: Constant integration resolving

Evaluate function boundary conditions at x = √(2)$x = \sqrt{2}$:

Given 5f(√(2)) = -4 f(√(2)) = -(4)/(5)$5f(\sqrt{2}) = -4 \implies f(\sqrt{2}) = -\frac{4}{5}$.

Step 2: Numeric tracking value

Evaluate final targeted definition state value at x=1$x=1$:

Pattern Recognition

Splitting powers of x$x$ to create a direct match with internal derivative differential flags speeds up the integration transformation sequence.

Chapter Mix

Class 12 Mathematics: Integrals

Related Formula

Domain of log chain iterations: For ₂ ₄(M) > 0$\log_2\log_4(M) > 0$, we require ₄(M) > 1 M > 4$\log_4(M) > 1 \implies M > 4$.

Core Logic

Trace internal arguments outward sequentially:

Thus, determine limits:

Step 1: Setting up the greatest integer function integration

We need to evaluate ∫₀² [x²] dx$\int_0^2 [x^2] \, \mathrm{d}x$. Identify step boundary switch locations inside range [0, 2]$[0, 2]$:

• For x in [0, 1): [x²] = 0$x \in [0, 1): [x^2] = 0$
• For x in [1, √(2)): [x²] = 1$x \in [1, \sqrt{2}): [x^2] = 1$
• For x in [√(2), √(3)): [x²] = 2$x \in [\sqrt{2}, \sqrt{3}): [x^2] = 2$
• For x in [√(3), 2): [x²] = 3$x \in [\sqrt{3}, 2): [x^2] = 3$

Set up separate boundary component integrations:

Step 2: Matching coefficients

Compare values with requested answer template shape:

Pattern Recognition

Integrals over greatest integer configurations change value exactly where the inner expression tracks through integer milestones. Mapping boundaries accurately resolves calculations smoothly.

Chapter Mix

Class 12 Mathematics: Integrals