Let f: R → R$f:\mathbb{R} \to \mathbb{R}$ be a function defined f(x) = x(1 + x⁴)1/4$f(x) = \frac{x}{(1 + x^4)^{1/4}}$ and g(x) = f(f(f(f(x))))$g(x) = f(f(f(f(x))))$. Then 18∫₀^ 2√(5) x² g(x) dx$18\int_{0}^{\sqrt{2\sqrt{5}}} x^2 g(x) dx$ is

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:

Alternatively, splitting the boundary metrics via continuous definite limits yields identical whole tracking blocks matching exactly to 12 total units.

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