The value of ∫₋₁¹ (1 + √(|x| - x))e^x + (√(|x| - x))e-xe^x + e-x dx$\int_{-1}^{1}\frac{\left(1 + \sqrt{|x| - x}\right)e^x + \left(\sqrt{|x| - x}\right)e^{-x}}{e^x + e^{-x}} \, \mathrm{d}x$ 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 [cite: 1361, 1362]: 3 - x² = t² x dx = -t dt$$3 - x^2 = t^2 \implies x \, \mathrm{d}x = -t \, \mathrm{d}t$$ [cite: 1361, 1362] Rewrite the internal integral block components [cite: 1363]: f(x) = ∫ x² · √(3-x²) · (x dx) = ∫ (3-t²) · t · (-t dt)$$f(x) = \int x^2 \cdot \sqrt{3-x^2} \cdot (x \, \mathrm{d}x) = \int (3-t^2) \cdot t \cdot (-t \, \mathrm{d}t)$$ [cite: 1363] = ∫ (t⁴ - 3t²) dt = (t⁵)/(5) - t³ + C$$= \int (t^4 - 3t^2) \, \mathrm{d}t = \frac{t^5}{5} - t^3 + C$$ [cite: 1363, 1366]

Return to original reference variable x$x$ [cite: 1366]: f(x) = (3-x²)5/25 - (3-x²)3/2 + C$$f(x) = \frac{(3-x^2)^{5/2}}{5} - (3-x^2)^{3/2} + C$$ [cite: 1366]

Step 1: Constant integration resolving

Evaluate function boundary conditions at x = √(2)$x = \sqrt{2}$ [cite: 1366]: f(√(2)) = (3-2)5/25 - (3-2)3/2 + C = (1)/(5) - 1 + C = -(4)/(5) + C$$f(\sqrt{2}) = \frac{(3-2)^{5/2}}{5} - (3-2)^{3/2} + C = \frac{1}{5} - 1 + C = -\frac{4}{5} + C$$ [cite: 1366] Given 5f(√(2)) = -4 f(√(2)) = -(4)/(5)$5f(\sqrt{2}) = -4 \implies f(\sqrt{2}) = -\frac{4}{5}$ [cite: 638, 1366]. -(4)/(5) + C = -(4)/(5) C = 0$$-\frac{4}{5} + C = -\frac{4}{5} \implies C = 0$$ [cite: 1366]

Step 2: Numeric tracking value

Evaluate final targeted definition state value at x=1$x=1$ [cite: 1367]: f(1) = (3-1)5/25 - (3-1)3/2 = 25/25 - 23/2$$f(1) = \frac{(3-1)^{5/2}}{5} - (3-1)^{3/2} = \frac{2^{5/2}}{5} - 2^{3/2}$$ [cite: 1367] = 23/2((2)/(5) - 1) = 2√(2)(-(3)/(5)) = - 6√(2)5$$= 2^{3/2}\left(\frac{2}{5} - 1\right) = 2\sqrt{2}\left(-\frac{3}{5}\right) = -\frac{6\sqrt{2}}{5}$$ [cite: 1367, 1368]

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 [cite: 1393, 1394]: ₄ ₆(3 + 4x - x²) > 0 ₆(3 + 4x - x²) > 1$$\log_{4}\log_{6}(3 + 4x - x^2) > 0 \implies \log_{6}(3 + 4x - x^2) > 1$$ [cite: 1393, 1394] 3 + 4x - x² > 6¹ x² - 4x + 3 < 0$$3 + 4x - x^2 > 6^1 \implies x^2 - 4x + 3 < 0$$ [cite: 1395, 1396] (x-1)(x-3) < 0 x in (1, 3)$$(x-1)(x-3) < 0 \implies x \in (1, 3)$$ [cite: 1397, 1398]

Thus, determine limits [cite: 1399]: a = 1, b = 3 b - a = 2$$a = 1, \quad b = 3 \implies b - a = 2$$ [cite: 1399]

Step 1: Setting up the greatest integer function integration

We need to evaluate ∫₀² [x²] dx$\int_0^2 [x^2] \, \mathrm{d}x$[cite: 1400]. Identify step boundary switch locations inside range [0, 2]$[0, 2]$ [cite: 1400]:

• 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 [cite: 1400]: ∫₀² [x²] dx = ∫₀¹ 0 dx + ∫₁√(2) 1 dx + ∫√(2)√(3) 2 dx + ∫√(3)² 3 dx$$\int_0^2 [x^2] \, \mathrm{d}x = \int_0^1 0 \, \mathrm{d}x + \int_1^{\sqrt{2}} 1 \, \mathrm{d}x + \int_{\sqrt{2}}^{\sqrt{3}} 2 \, \mathrm{d}x + \int_{\sqrt{3}}^2 3 \, \mathrm{d}x$$ [cite: 1400] = 0 + (√(2) - 1) + 2(√(3) - √(2)) + 3(2 - √(3))$$= 0 + (\sqrt{2} - 1) + 2(\sqrt{3} - \sqrt{2}) + 3(2 - \sqrt{3})$$ [cite: 1400] = √(2) - 1 + 2√(3) - 2√(2) + 6 - 3√(3) = 5 - √(2) - √(3)$$= \sqrt{2} - 1 + 2\sqrt{3} - 2\sqrt{2} + 6 - 3\sqrt{3} = 5 - \sqrt{2} - \sqrt{3}$$ [cite: 1400]

Step 2: Matching coefficients

Compare values with requested answer template shape [cite: 1400]: 5 - √(2) - √(3) = p - √(q) - √(r)$$5 - \sqrt{2} - \sqrt{3} = p - \sqrt{q} - \sqrt{r}$$ [cite: 1400] p = 5, q = 2, r = 3$$p = 5, \quad q = 2, \quad r = 3$$ [cite: 1400]

Final Sum = p + q + r = 5 + 2 + 3 = 10$$\text{Final Sum} = p + q + r = 5 + 2 + 3 = 10$$ [cite: 1400]

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