Arctans

Calculus Level 3

∫ 0 1 arctan ⁡ x d x = π − a ln ⁡ a b \large \int_0^1 \arctan x \, dx = \dfrac{\pi - a \ln a}b

If the equation above holds true for integers a a and b b , find a + b a+b .


The answer is 6.

Ashish Menon by Ashish Menon , 20, India

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1 solution

Ashish Menon
May 21, 2016

∫ 0 1 arctan ⁡ ( x ) d x = x arctan ⁡ ( x ) ∣ 0 1 − ∫ 0 1 x x 2 + 1 d x = 1 arctan ⁡ ( 1 ) − 0 arctan ⁡ ( 0 ) − ∫ 0 1 x x 2 + 1 d x = π 4 − ln ⁡ ( 2 ) 2 = π − 2 ln ⁡ ( 2 ) 4 ∴ a + b = 2 + 4 = 6 \begin{aligned} \int_{0}^{1} \arctan(x) \ dx & = x\arctan(x){\Huge{\vert}}_{0}^{1} - \int_{0}^{1} \dfrac{x}{x^2 + 1} \ dx\\ \\ & = 1\arctan(1) - 0\arctan(0) - \int_{0}^{1} \dfrac{x}{x^2 + 1} \ dx\\ \\ & = \dfrac{\pi}{4} - \dfrac{\ln(2)}{2}\\ \\ & = \dfrac{\pi - 2\ln(2)}{4}\\ \\ \therefore a + b & = 2 + 4\\ & = \boxed{6} \end{aligned}

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where is integer b b in the problem ?

Sabhrant Sachan - 5 years ago

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Someone edited my problem, so it is changed. I edited it now.

Ashish Menon - 5 years ago

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