A calculus problem by U Z

Calculus Level 3

Which of the following is an antiderivative of ∫ d x ( 2 x − 7 ) ( x 2 − 7 x + 12 ) 1 / 2 \displaystyle \int \dfrac {dx}{(2x-7)(x^2 - 7x+12)^{1/2}} ?

s e c − 1 ( 2 x + 7 ) sec^{-1}( 2x + 7) 1 2 s e c − 1 ( 2 x − 7 ) \frac{1}{2}sec^{-1}( 2x - 7) s e c − 1 ( 2 x − 7 ) sec^{-1}( 2x - 7) 2 s e c − 1 ( 2 x − 7 ) 2sec^{-1}( 2x - 7)

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U Z by U Z , 24, Guyana

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1 solution

U Z
Oct 10, 2014

multiplying and dividing by 2

∫ 2 ( 2 x − 7 ) ( ( 4 x 2 − 28 x + 48 ) 1 2 ) \int\ \frac{2}{ (2x - 7)((4x^{2} - 28x +48)^\frac{1}{2})}

thus,

∫ 2 ( 2 x − 7 ) ( ( 4 x 2 − 2.2.7 x + 49 − 1 ) 1 2 ) \int\ \frac{2}{ (2x - 7)((4x^{2} - 2.2.7x +49 - 1)^\frac{1}{2})}

∫ 2 ( 2 x − 7 ) ( ( 2 x − 7 ) 2 − 1 ) 1 2 ) \int\ \frac{2}{ (2x - 7)((2x - 7)^{2} - 1)^\frac{1}{2})}

2x -7 =t

thus it becomes the derivative of s e c − 1 t sec^{-1}t ( fundamental formula)

thus answer is s e c − 1 ( 2 x − 7 ) sec^{-1}(2x - 7)

7 Helpful 0 Interesting 0 Brilliant 0 Confused

very good solution thanks

sunny malhotra - 6 years, 6 months ago

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