A nested integral

Calculus Level 4

Let F ( f ( x ) ) F(f(x)) denote ∫ 0 x f ( s ) d s \int_0 ^ x f(s) \, ds .

If the algebraic expression of x 0 ⋅ F ( x 1 ⋯ F ( x 8 ⋅ ( F ( x 9 ⋅ F ( x 10 ) ) ) ) ⋯ ) x^0 \cdot F(x^1 \cdots F(x^8 \cdot (F(x^9 \cdot F(x^{10} )))) \cdots ) is equal to x A B \dfrac {x^A}{B} , what is the value of ( A + B ) ( m o d 1 0 10 ) (A+B) \pmod {10^{10}} .


The answer is 3035760065.

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Roel Baars by Roel Baars , 33, Netherlands

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1 solution

Lu Chee Ket
Dec 10, 2015

To write in short by presuming readers know how to integrate but just to check with this,

x 65 11 × 21 × 30 × 38 × 45 × 51 × 56 × 60 × 63 × 65 \displaystyle \frac{x^{65}}{11 \times 21 \times 30 \times 38 \times 45 \times 51 \times 56 \times 60 \times 63 \times 65}

A + B = 65 + 8315583035760000 = 8315583035760065

8315583035760065 MOD 10000000000 = 3035760065

Answer: 3035760065 \boxed{3035760065}

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