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Question: The value of \(\int _ { 1 } ^ { 16 } \tan ^ { - 1 }\)![](https://cdn.pureessence.tech/canvas_555.png...

The value of 116tan1\int _ { 1 } ^ { 16 } \tan ^ { - 1 }dx is -

A

3\sqrt { 3 }

B

43\frac { 4 } { 3 }p – 23\sqrt { 3 }

C

43\frac { 4 } { 3 }p + 23\sqrt { 3 }

D

163\frac { 16 } { 3 }p – 23\sqrt { 3 }

Answer

163\frac { 16 } { 3 }p – 23\sqrt { 3 }

Explanation

Solution

Integrating by parts, the given integral is equal to

x tan–1 14xx1\frac { 1 } { 4 \sqrt { x } \sqrt { \sqrt { x } - 1 } }dx

= 163\frac { 16 } { 3 }p – 14\frac { 1 } { 4 }

= 163\frac { 16 } { 3 }p – 14\frac { 1 } { 4 }dt (= 1 + t2)

= 163\frac { 16 } { 3 }p – (3\sqrt { 3 } + 16π3\frac { 16 \pi } { 3 } – 2 3\sqrt { 3 } .