Definite Integration Question - None

Question

If $\displaystyle\int_0^{\pi} \frac{\sin x(\sin x + 1)e^{\sin x + \cos x}}{e^{\cos x} + 1}\, dx = a + b\int_0^{\pi} e^{\sin x}\, dx$ where $a$ and $b$ are positive rational numbers, then find the value of $100(a^2 + b^2)$.

Accepted Answer

Step 1: Let I = ∫₀^π [sin x(sin x + 1)e^(sin x + cos x)]/[e^(cos x) + 1] dx

Step 2: Split the numerator: sin x(sin x + 1)e^(sin x + cos x) = sin x·sin x·e^(sin x + cos x) + sin x·e^(sin x + cos x)

Step 3: For the second part, use substitution property. Consider I = ∫₀^π [sin x·e^(sin x + cos x)]/[e^(cos x) + 1] · (sin x + 1) dx

Step 4: Apply the property: ∫₀^π f(x)dx + ∫₀^π f(π-x)dx. Note that at x and π-x: sin(π-x) = sin x and cos(π-x) = -cos x

Step 5: When we compute I + I' (where I' uses π-x substitution), the terms with denominator structure (e^(cos x) + 1) and (e^(-cos x) + 1) combine such that:

Step 6: After careful manipulation: I = 1 + ∫₀^π e^(sin x) dx · (1/2) can be derived by noting that [e^(sin x + cos x)]/(e^(cos x) + 1) + [e^(sin(π-x) + cos(π-x))]/(e^(cos(π-x)) + 1) = e^(sin x)

Step 7: This yields: 2I = 2·1 + 1·∫₀^π e^(sin x) dx, giving a = 1 and b = 1/2

Step 8: Therefore a² + b² = 1 + 1/4 = 5/4

∴ Answer: 100(a² + b²) = 100 × 5/4 = 125

Question:

Step-by-Step Solution

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