If α =   ∫ 0 1 e 9 x + 3 tan - 1 ⁡ x   12 + 9 x 2 1 + x 2   d x where tan - 1 ⁡ x   takes only principal values, then the value of log e ⁡ 1 + α -   3 π 4 is

Question

If &#945; = &#160; &#8747; 0 1 e 9 x + 3 tan - 1 &#8289; x &#160; 12 + 9 x 2 1 + x 2 &#160; d x where tan - 1 &#8289; x &#160; takes only principal values, then the value of log e &#8289; 1 + &#945; - &#160; 3 &#960; 4 is

Aicharya · Accepted Answer

&#945; = &#160; &#8747; 0 1 e 9 x + 3 tan - 1 &#8289; x &#160; 12 + 9 x 2 1 + x 2 &#160; d x = &#160; &#8747; 0 1 e 9 x + 3 tan - 1 &#8289; x &#160; 9 + 3 1 + x 2 &#160; d x = &#160; e 9 x + 3 tan - 1 &#8289; x | 0 1 = &#160; e 9 + 3 &#960; 4 - 1 &#8658; &#160;&#160; ln &#160; ( 1 + &#945; ) - &#160; 3 &#960; 4 = ln 1 + e 9 + 3 &#960; 4 - 1 - 3 &#960; 4 = ln e 9 + 3 &#960; 4 - 3 &#960; 4 = 9 .

If α =   ∫ 0 1 e 9 x + 3 tan - 1 ⁡ x   12 + 9 x 2 1 + x 2   d x where tan -…

Solution

If &#945; = &#160; &#8747; 0 1 e 9 x + 3 tan - 1 &#8289; x &#160; 12 + 9 x 2 1 + x 2 &#160; d x where tan -…

Solution

If α = ∫ 0 1 e 9 x + 3 tan - 1 ⁡ x 12 + 9 x 2 1 + x 2 d x where tan -…