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Integrals

Question

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If $integral subscript 0 superscript straight a fraction numerator 1 over denominator 4 plus straight x squared end fraction dx equals straight pi over 8$ , find the value of a.

Solution

bold Given bold space bold that integral subscript 0 superscript straight a fraction numerator dx over denominator 4 plus straight x squared end fraction equals straight pi over 8 We space need space to space find space the space value space of space straight a. Let space straight I space equals space integral subscript 0 superscript straight a fraction numerator dx over denominator 4 plus straight x squared end fraction equals space straight pi over 8 Thus comma space straight I space equals space 1 half open parentheses tan to the power of negative 1 end exponent straight x over 2 close parentheses subscript 0 superscript straight a equals space space straight pi over 8 rightwards double arrow 1 half tan to the power of negative 1 end exponent straight a over 2 equals straight pi over 8 rightwards double arrow tan to the power of negative 1 end exponent straight a over 2 equals 2 cross times straight pi over 8 rightwards double arrow tan to the power of negative 1 end exponent straight a over 2 equals straight pi over 4 rightwards double arrow straight a over 2 equals 1 rightwards double arrow straight a equals 2

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