Let f: Rarrow R$f:\mathbb{R}\rightarrow\mathbb{R}$ be a twice differentiable function such that f(2)=1$f(2)=1$. If F(x)=xf(x)$F(x)=xf(x)$ for all xin R$x\in\mathbb{R}$, ∫₀²xF(x)dx=6$\int_{0}^{2}xF^{\prime}(x)dx=6$ and ∫₀²x²F(x)dx=40$\int_{0}^{2}x^{2}F^{\prime\prime}(x)dx=40$, then F(2)+∫₀²F(x)dx$F^{\prime}(2)+\int_{0}^{2}F(x)dx$ is equal to:

Related Formula

Definite integration geometry area: Split boundary zones around intersections where functional dominant switches occur.

Core Logic

Analyze intersection points between y₁ = |x|$y_1 = |x|$ and y₂ = x|x-2|$y_2 = x|x-2|$ across required integration span regions:

• For x in [-2, 0]$x \in [-2, 0]$: |x| = -x$|x| = -x$ and x|x-2| = -x(2-x) = x² - 2x$x|x-2| = -x(2-x) = x^2 - 2x$. Max curve tracks through distinct segments.
• For positive sectors, compute intersections: x = x(2-x) x = 1$x = x(2-x) \implies x = 1$ or x=0$x=0$. Also check switch locations where graphs swap dominance.

Step 1: Setting up separate area integral blocks

Using geometric area partitions calculated across continuous regions [cite: 1495]: Area = (1)/(2) × 2 × 2 + (1)/(2) × 3 × 3 + (1)/(2) × 1 × 11 = 12$$\text{Area} = \frac{1}{2} \times 2 \times 2 + \frac{1}{2} \times 3 \times 3 + \frac{1}{2} \times 1 \times 11 = 12$$ [cite: 1495] Alternatively, splitting the boundary metrics via continuous definite limits yields identical whole tracking blocks matching exactly to 12 total units[cite: 1495].

Pattern Recognition

Plotting multiple curves dynamically highlights dominance shifts quickly. Computing distinct straight triangular chunks saves valuable integration time.

Chapter Mix

Class 12 Mathematics: Integrals (Application of Integrals)

Core Logic

We have two functional equations to solve before integrating.

Equation 1: f(x) + 2f((1)/(x)) = x² + 5$f(x) + 2f\left(\frac{1}{x}\right) = x^2 + 5$ Replace x$x$ with (1)/(x)$\frac{1}{x}$:

Multiplying this new equation by 2$2$ and subtracting the original Equation 1 eliminates the f((1)/(x))$f\left(\frac{1}{x}\right)$ term:

Step 1: Finding alpha

Integrate f(x)$f(x)$ from 1$1$ to 2$2$:

Thus, 9α = 11$9\alpha = 11$.

Step 2: Solving for g(x) and finding beta

We are given 2g(x) - 3g((1)/(2)) = x$2g(x) - 3g\left(\frac{1}{2}\right) = x$. Substitute x = (1)/(2)$x = \frac{1}{2}$:

Substitute this constant value back into the original equation:

Now find β$\beta$:

Step 3: Calculating 9alpha + beta

Combining our values:

Pattern Recognition

Functional equations involving x → (1)/(x)$x \to \frac{1}{x}$ are easily solved by treating the swapped forms as a system of linear equations, allowing direct isolation of the underlying function.

Chapter Mix

Class 12 Mathematics: Definite Integrals Class 12 Mathematics: Functional Equations

Core Logic

Let's simplify the integrand by rationalizing the denominator term block. Notice that:

Substituting this back into the denominator expression column:

Step 1: Implementing the Substitution Path

Let t = √(1+x²) + x$t = \sqrt{1+x^2} + x$. Then:

Since √(1+x²) + x = t$\sqrt{1+x^2} + x = t$ and √(1+x²) - x = (1)/(t)$\sqrt{1+x^2} - x = \frac{1}{t}$, adding both gives:

Thus, dx = (1)/(2t)(t + (1)/(t)) dt = (1)/(2)(1 + (1)/(t²)) dt$dx = \frac{1}{2t}\left(t + \frac{1}{t}\right) dt = \frac{1}{2}\left(1 + \frac{1}{t^2}\right) dt$.

Step 2: Integrating with respect to t

Substitute these back into the integral:

Step 3: Matching Form and Finding m + n

To match the template format, let's pull out a factor of t$t$:

Comparing this directly with the given answer format, we identify:

• m = 360$m = 360$
• n = 19$n = 19$

Computing m + n$m + n$:

Pattern Recognition

Expressions containing conjugate factors like √(1+x²) ± x$\sqrt{1+x^2} \pm x$ frequently simplify under rationalization because their product equals 1. This dynamic quickly reduces fractional components into single power blocks.

Chapter Mix

Class 12 Mathematics: Indefinite Integrals

Related Formula

King's property of definite integrals:

Core Logic

Let the given integral be I$I$. Apply King's property by substituting x → -x$x \to -x$:

Add both integral expressions 2I = I + I$2I = I + I$:

Step 1: Apply Symmetry Properties

The integrand is completely even. Hence, convert intervals:

For x in [0,1]$x \in [0,1]$, |x| = x √(|x| - x) = 0$|x| = x \implies \sqrt{|x| - x} = 0$ and √(|x| + x) = √(2x)$\sqrt{|x| + x} = \sqrt{2x}$:

Step 2: Final Integration Execution

Pattern Recognition

When functions involve combinations of exponential components (e^x, e-x$e^x, e^{-x}$) over symmetric boundaries, adding the variable reflection eliminates exponential fractions instantly.

Chapter Mix

Class 12 Mathematics: Definite Integration

Related Formula

The area enclosed between upper bounding function yupper$y_{\text{upper}}$ and lower function ylower$y_{\text{lower}}$ is:

Core Logic

We need to find the intersection points of the curves to understand the x+7, 11-3x$\min\{x+7, 11-3x\}$ boundary transition:

• x+7 = 11-3x 4x = 4 x = 1$x+7 = 11-3x \implies 4x = 4 \implies x = 1$.

Hence, the line switches behavior at x=1$x=1$.

• Intersecting 1+x²$1+x^2$ with x+7$x+7$:

• Intersecting 1+x²$1+x^2$ with 11-3x$11-3x$:

Step 1: Set up Integrals

The transition points show that from x = -2$x = -2$ to 1$1$, the upper line is x+7$x+7$, and from x = 1$x = 1$ to 2$2$, the upper line is 11-3x$11-3x$.

Step 2: Integration Evaluation

Evaluating the first integral:

Evaluating the second integral:

Total Area A$A$:

Step 3: Calculate 3A

Multiplying the total area by 3:

Pattern Recognition

When a boundary contains a $\min\{\}$ or $\max\{\}$ component, always solve for their internal intersection first to identify the exact splitting point of your definite integrals.

Chapter Mix

Class 12 Mathematics: Integral Calculus