A value of α such that ∫ α α + 1 d x ( x + α ) ( x + α + 1 ) = l o g e 9 8 is

Question

A value of &#945; such that &#8747; &#945; &#945; + 1 d x ( x + &#945; ) ( x + &#945; + 1 ) = l o g e 9 8 is

Aicharya · Accepted Answer

I = &#8747; &#945; &#945; + 1 d x ( x + &#945; ) ( x + &#945; + 1 ) = &#8747; &#945; &#945; + 1 1 x + &#945; &#160; &#8211; 1 x + &#945; + 1 &#160; d x Using partial fractions = l n x + &#945; - l n x + &#945; + 1 &#945; &#945; + 1 = l n x + &#945; x + &#945; + 1 &#160; &#945; &#945; + 1 = l n 2 &#945; + 1 2 &#945; + 2 - l n 2 &#945; 2 &#945; + 1 = l n 2 &#945; + 1 2 2 &#945; + 1 2 - 1 = ln &#8289; 9 8 &#8658; ( 2 &#945; + 1 ) 2 = 9 &#8658; 2 &#945; + 1 = &#177; 3 &#8658; &#945; = 1 or - 2 .

A value of &#945; such that &#8747; &#945; &#945; + 1 d x ( x + &#945; ) ( x + &#945; + 1 ) = l o g e 9 8 is

Solution

A value of α such that ∫ α α + 1 d x ( x + α ) ( x + α + 1 ) = l o g e 9 8 is