Definite integral

Calculus Level 5

If

∫ − 1 3 1 3 x 4 1 − x 4 cos ⁡ − 1 ( 2 x 1 + x 2 ) d x \displaystyle\int _{ -\frac { 1 }{ \sqrt { 3 } } }^{ \frac { 1 }{ \sqrt { 3 } } }{ \frac { { x }^{ 4 } }{ 1-{ x }^{ 4 } } } \cos ^{ -1 } \left({ \frac { 2x }{ 1+{ x }^{ 2 } } } \right) dx can be represented in the form π a [ π + b log ⁡ ( c + 3 ) + d 3 ] \frac { \pi }{ a } [\pi +b\log { (c+\sqrt { 3 }) } +d\sqrt { 3 } ] then find the value of a + b + c + d a+b+c+d .


The answer is 13.

Abdulmuttalib Lokhandwala by abdulmuttalib lokhandwala , 23

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1 solution

cos ⁡ − 1 y = π 2 − sin ⁡ − 1 y cos ⁡ − 1 2 x 1 + x 2 = π 2 − sin ⁡ − 1 2 x 1 + x 2 = π 2 − 2 tan ⁡ − 1 x I = ∫ − 1 3 1 3 [ π 2 X 4 1 − X 4 − X 4 1 − X 4 2 tan ⁡ − 1 x ] d x A s X 4 1 − X 4 2 tan ⁡ − 1 x i s a n o d d f u n c t i o n i t s i n t e g r a l w i l l b e z e r o I = 2 π 2 ∫ 0 1 3 [ − 1 + 1 1 − x 4 ] d x S o l v i n g t h i s w e g e t I = π 12 [ π + 3 log ⁡ ( 2 + 3 ) − 4 3 ] T h e r e f o r e a + b + c + d = 13 \cos ^{ -1 }{ y } =\quad \frac { \pi }{ 2 } -\sin ^{ -1 }{ y } \\ \cos ^{ -1 }{ \frac { 2x }{ 1+{ x }^{ 2 } } } =\frac { \pi }{ 2 } -\sin ^{ -1 }{ \frac { 2x }{ 1+{ x }^{ 2 } } } \\ \quad \quad \ \quad \quad =\frac { \pi }{ 2 } -2\tan ^{ -1 }{ x } \quad \\ I\quad =\quad \int _{ -\frac { 1 }{ \sqrt { 3 } } }^{ \frac { 1 }{ \sqrt { 3 } } }{ [\frac { \pi }{ 2 } \frac { { X }^{ 4 } }{ 1-{ X }^{ 4 } } } -{ \frac { { X }^{ 4 } }{ 1-{ X }^{ 4 } } }2\tan ^{ -1 }{ x } ]dx\\ As\quad { \frac { { X }^{ 4 } }{ 1-{ X }^{ 4 } } }2\tan ^{ -1 }{ x }\quad is\quad an\quad odd\quad function\quad its\quad integral\quad will\quad be\quad zero\\ I\quad =2\frac { \pi }{ 2 } \int _{ 0 }^{ \frac { 1 }{ \sqrt { 3 } } }{ [-1+\frac { 1 }{ 1-{ x }^{ 4 } } } ]dx\\ \\ Solving\quad this\quad we\quad get\quad I\quad =\quad \frac { \pi }{ 12 } [\pi +3\log {( 2+\sqrt { 3 } ) } -4\sqrt { 3 } ]\\ Therefore\quad a+b+c+d=13\\ \\ \\ \\ \\

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I solved it in the exact same manner

Ishan Shah - 6 years, 11 months ago

I don't understand the third line, which property did you used? Thanks! Nice solution!

Carlos David Nexans - 6 years, 10 months ago

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put x equal to tan(theta/2)

Himanshu Aswani - 5 years, 5 months ago

Superb solution

Sudhamsh Suraj - 4 years, 7 months ago

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