Limit of Definite Integral #1
I n = ∫ 0 1 a x + b x n d x
Define the integral I n as above for positive real variables a and b independent of x and natural number n .
n → ∞ lim n I n = λ a + μ b 1
If λ and μ are constants that satisfy the limit above, evaluate λ + μ .
The answer is 2.
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2 solutions
I n n I n n → ∞ lim n I n = ∫ 0 1 a x + b x n d x = n 1 ∫ 0 1 a t n 1 + b t n 1 d t = ∫ 0 1 a t n 1 + b t n 1 d t a s n → ∞ t n 1 → 1 = ∫ 0 1 a + b 1 d t = a + b 1 λ = 1 , μ = 1 x = t n 1 , d x = n 1 t n 1 − 1