Let f(x) + 2f((1)/(x)) = x² + 5$f(x) + 2f\left(\frac{1}{x}\right) = x^2 + 5$ and 2 g (x) - 3 g ((1)/(2)) = x, x > 0$2 \mathrm {g} (\mathrm {x}) - 3 \mathrm {g} \left(\frac {1}{2}\right) = \mathrm {x}, \mathrm {x} > 0$. If α = ∫_ 1 ^ 2 f (x) d x$\alpha = \int_ {1} ^ {2} f (x) d x$, and β = ∫_ 1 ^ 2 g (x) d x$\beta = \int_ {1} ^ {2} g (x) d x$, then the value of 9α + β$9\alpha + \beta$ is:

Related Formula

King's Property for symmetric integration limits:

Core Logic

Let the given definite integral be I$I$:

Apply King's property by replacing x$x$ with -x$-x$ (since -π/2 + π/2 = 0$-\pi/2 + \pi/2 = 0$):

Step 1: Simplify by Adding Expressions

Adding equations (1) and (2):

Since the integrand is even, we can change the limits from 0$0$ to π/2$\pi/2$:

Step 2: Substitution and Algebraic Deconstruction

Let x = t x dx = dt$\sin x = t \implies \cos x \, dx = dt$. Limits change from 0$0$ to 1$1$:

Dividing the numerator and denominator by t²$t^2$, we write it as two distinct expressions:

where z = t - (1)/(t)$z = t - \frac{1}{t}$ and k = t + (1)/(t)$k = t + \frac{1}{t}$.

Step 3: Integrate and Evaluate Parameters

Evaluating standard anti-derivatives:

Matching parameters yields α = 2$\alpha = 2$ and β = 2$\beta = 2$. Thus:

Pattern Recognition

Sees: Exponential variables causing asymmetry in symmetric integration bounds. Shortcut: Using King's rule completely eliminates the confusing ex$e^{\sin x}$ factor, leaving behind a straightforward rational trigonometric configuration.

Chapter Mix

Class 12 Mathematics: Definite Integrals

Related Formula

Core Logic

Let us partition the given integral I$I$ into two clean parts I₁$I_1$ and I₂$I_2$:

Factoring out appropriate powers of x$\sin x$ and x$\cos x$ from inside the roots:

Step 1: Evaluating the Integrals

For I₁$I_1$, substitute x θ - θ = t² ² x dx = (2t dt)/( θ)$\tan x \cos \theta - \sin \theta = t^2 \implies \sec^2 x \, dx = \frac{2t \, dt}{\cos \theta}$:

For I₂$I_2$, substitute θ - x θ = z² ² x dx = (2z dz)/( θ)$\cos \theta - \cot x \sin \theta = z^2 \implies \csc^2 x \, dx = \frac{2z \, dz}{\sin \theta}$:

Step 2: Combining Coefficients

Comparing with the given expression, we find the coefficients:

Multiplying them together:

Pattern Recognition

When integrating roots of trigonometric functions involving (x - θ)$(x - \theta)$, try dividing/multiplying fields by ⁿ x$\cos^n x$ or ⁿ x$\sin^n x$ to force x / ² x$\tan x / \sec^2 x$ templates.

Chapter Mix

Class 12 Mathematics: Integrals

Related Formula

Core Logic

The absolute function | x - x|$|\cos x - \sin x|$ flips sign at x = (π)/(4)$x = \frac{\pi}{4}$ within the boundary range:

• For x in [(π)/(6), (π)/(4)]$x \in \left[\frac{\pi}{6}, \frac{\pi}{4}\right]$: x ≥ x | x - x| = x - x$\cos x \geq \sin x \implies |\cos x - \sin x| = \cos x - \sin x$
• For x in [(π)/(4), (π)/(3)]$x \in \left[\frac{\pi}{4}, \frac{\pi}{3}\right]$: x ≥ x | x - x| = x - x$\sin x \geq \cos x \implies |\cos x - \sin x| = \sin x - \cos x$

Step 1: Boundary Splitting Integration

Step 2: Coefficient Matrix Matching

Comparing with the given baseline layout:

Evaluating the required target metrics:

Pattern Recognition

Never forget that √(f(x)²) = |f(x)|$\sqrt{f(x)^2} = |f(x)|$. Skipping modulus checks inside definite root boundaries leads to incorrect answers.

Chapter Mix

Class 12 Mathematics: Integrals

Related Formula

Core Logic

First, find the orthocentre (a,b)$(a,b)$ of Δ ABC$\Delta ABC$ with vertices A(1, 2)$A(1, 2)$, B(2, 3)$B(2, 3)$, and C(3, 1)$C(3, 1)$. Slope of AB = (3-2)/(2-1) = 1$AB = \frac{3-2}{2-1} = 1$. The altitude from C$C$ onto AB$AB$ must be perpendicular to AB$AB$, so its slope is -1$-1$. Equation of altitude from C(3,1)$C(3,1)$:

The orthocentre (a, b)$(a, b)$ lies on all altitudes, including this one. Thus, it satisfies a + b = 4$a + b = 4$.

Step 1: Applying Definite Integral Properties

Given I₁ = ∫ₐ^b x (4x-x²) dx$I_1 = \int_a^b x \sin(4x-x^2) dx$, let's rewrite the argument of sine:

Apply King's Rule replacing x$x$ with (a+b-x)$(a+b-x)$. Since we proved a+b = 4$a+b = 4$, substitute x$x$ with (4-x)$(4-x)$:

Step 2: Evaluating the Integral Ratio

Expand the newly formed integral:

Notice that the second term is I₁$I_1$ and the first integral is I₂$I_2$:

Step 3: Final Output Evaluation

We need the value of 36 (I₁)/(I₂)$36 \frac{I_1}{I_2}$:

Pattern Recognition

Whenever you see ∫ₐ^b x · f(x(a+b-x)) dx$\int_a^b x \cdot f(x(a+b-x)) dx$, immediately apply King's Rule to factor out x$x$. You rarely need the individual values of the integration bounds, only their sum.

Chapter Mix

Class 11 Maths: Straight Lines Class 12 Maths: Definite Integration

Related Formula

Core Logic

To evaluate integrals with sum of square roots in the denominator, multiply and divide by the conjugate to rationalize it.

Step 1: Integration and Bounds Setup

Integrate the resulting expression:

Step 2: Term Simplification

Evaluate the boundary powers: 43/2 = 8$4^{3/2} = 8$ 23/2 = 2√(2)$2^{3/2} = 2\sqrt{2}$ 33/2 = 3√(3)$3^{3/2} = 3\sqrt{3}$ 13/2 = 1$1^{3/2} = 1$

Substitute back into the expression:

Step 3: Finding Co-efficients

Comparing with a+b√(2)+c√(3)$a+b\sqrt{2}+c\sqrt{3}$ yields: a = 3$a = 3$, b = -(2)/(3)$b = -\frac{2}{3}$, c = -1$c = -1$

Compute 2a+3b-4c$2a+3b-4c$:

6 - 2 + 4 = 8$6 - 2 + 4 = 8$

Pattern Recognition

Whenever you see a sum of square roots in the denominator of an integrand, the immediate algorithmic next step is rationalization.

Chapter Mix

Class 12 Maths: Definite Integration