A Strange Arctangent

Calculus Level pending

∫ 1 2 1 arctan ⁡ ( 2 x − 1 ) d x = π − ln ⁡ A B \int_{\frac{1}{2}}^{1}\arctan(2x-1)\text{ }dx=\dfrac{\pi-\ln A}{B} What is A + B A+B ?


The answer is 12.

Trevor B. by Trevor B. , 22, USA

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1 solution

Milo Štěpán
May 11, 2014

The way I did it (not sure if it's the best though) Let: 2 x − 1 = a 2x-1=a d x = d a 2 dx=\frac { da }{ 2 }

Now we use known integral: ∫ arctan ⁡ a ⋅ d a = a ⋅ arctan ⁡ a − ln ⁡ ( 1 + a 2 ) 2 \int { \arctan ^{ }{ a } \cdot } da=a\cdot \arctan ^{ }{ a } -\frac { \ln { (1+{ a }^{ 2 }) } }{ 2 }

Now revert substitution and plug in values to get π − ln ⁡ 4 8 \frac { \pi -\ln { 4 } }{ 8 }

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