Interesting Integral

Calculus Level pending

The value of

e ∫ − 1 1 15 x − 25 4 x 2 + 15 x − 25 d x e^{\int^1_{-1} \frac{15x-25}{4x^2+15x-25}dx}

Can be expressed as a b c \frac{a\sqrt{b}}{c} where a , b , c a, b, c are integers, a , c a, c are relatively prime, and b b is not divisible by any square number greater than 1. Find a+b+c.


The answer is 100.

Boaz Guberman by Boaz Guberman , 21, Israel

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1 solution

Solve the integration first,

I = ∫ − 1 1 15 x − 25 4 x 2 + 15 x − 25 d x = ∫ − 1 1 ( 4 x + 5 − 1 4 x − 5 ) d x = 4 ln ⁡ ∣ x + 5 ∣ − 1 4 ln ⁡ ∣ 4 x − 5 ∣ ∣ − 1 1 = ln ⁡ ( 81 3 16 ) \displaystyle \begin{aligned} I &= \int_{-1}^{1}\frac{15x-25}{4x^2+15x-25} \, \mathrm{d}x \\ &= \int_{-1}^{1} \left( \frac{4}{x+5} - \frac{1}{4x-5} \right) \, \mathrm{d} x \\ &= \left. 4 \ln |x+5| - \frac{1}{4} \ln |4x-5| \right|_{-1}^{1} \\ &= \ln \left( \frac{81 \sqrt{3}}{16} \right) \end{aligned}

Hence, e I = 81 3 16 \displaystyle \mathrm{e}^{I} = \frac{81 \sqrt{3}}{16} imply a + b + c = 100 \displaystyle a+b+c=\boxed{100}

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