Q28
jee_main_2024_01_february_morning
Properties of Definite Integrals
If int_-pi/2^pi/2frac8sqrt2cos x\,dx(1+e^sin x)(1+sin^4x)=alphapi+beta log_e(3+2sqrt2), where alpha, beta are integers, then alpha^2+beta^2 equals
Numerical Answer. Answer: 8 to 8
Solution
### Related Formula
King's Property for symmetric integration limits:
int_-a^a f(x) \, dx = int_-a^a f(-x) \, dx
### Core Logic
Let the given definite integral be I:
I = int_-pi/2^pi/2frac8sqrt2cos x\,dx(1+e^sin x)(1+sin^4x) quad implies (1)
Apply King's property by replacing x with -x (since -pi/2 + pi/2 = 0):
I = int_-pi/2^pi/2frac8sqrt2cos(-x)\,dx(1+e^sin(-x))(1+sin^4(-x))
I = int_-pi/2^pi/2frac8sqrt2cos x\,dx(1+e^-sin x)(1+sin^4x) = int_-pi/2^pi/2frac8sqrt2cos x cdot e^sin x\,dx(e^sin x+1)(1+sin^4x) quad implies (2)
### Step 1: Simplify by Adding Expressions
Adding equations (1) and (2):
2I = int_-pi/2^pi/2 frac8sqrt2cos x(1 + e^sin x)\,dx(1+e^sin x)(1+sin^4x)
2I = int_-pi/2^pi/2 frac8sqrt2cos x\,dx1+sin^4x
Since the integrand is even, we can change the limits from 0 to pi/2:
2I = 2 int_0^pi/2 frac8sqrt2cos x\,dx1+sin^4x implies I = int_0^pi/2 frac8sqrt2cos x\,dx1+sin^4x
### Step 2: Substitution and Algebraic Deconstruction
Let sin x = t implies cos x \, dx = dt. Limits change from 0 to 1:
I = int_0^1 frac8sqrt2\,dt1+t^4 = 4sqrt2 int_0^1 frac2\,dt1+t^4
Dividing the numerator and denominator by t^2, we write it as two distinct expressions:
I = 4sqrt2 left[ int_0^1 frac1+frac1t^2t^2+frac1t^2 \, dt - int_0^1 frac1-frac1t^2t^2+frac1t^2 \, dt right]
I = 4sqrt2 left[ int_-infty^0 fracdzz^2+2 - int_infty^2 fracdkk^2-2 right]
where z = t - frac1t and k = t + frac1t.
### Step 3: Integrate and Evaluate Parameters
Evaluating standard anti-derivatives:
I = 4sqrt2 left[ frac1sqrt2tan^-1left(fraczsqrt2right) right]_-infty^0 - 4sqrt2 left[ frac12sqrt2lnleft|frack-sqrt2k+sqrt2right| right]_infty^2
I = 4sqrt2left(fracpi2sqrt2right) - 2 lnleft|frac2-sqrt22+sqrt2right| = 2pi - 2ln(sqrt2-1)^2
I = 2pi + 2ln(3+2sqrt2)
Matching parameters yields alpha = 2 and beta = 2. Thus:
alpha^2 + beta^2 = 2^2 + 2^2 = 8
### Pattern Recognition
Sees: Exponential variables causing asymmetry in symmetric integration bounds.
Shortcut: Using King's rule completely eliminates the confusing e^sin x factor, leaving behind a straightforward rational trigonometric configuration.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 12 Mathematics: Definite Integrals
