Small looking integrals have really big solutions

Calculus Level 3

$\large \int _{ 0 }^{ 1 }{ \frac { { ( 1 -16{ x }^{ \frac { 1 }{ 4 } } ) }^{ 2 } }{ { ( { 2 }^{ 8 }\ x ) }^{ \frac { 7 }{ 8 } } } \, dx }$

If the integral above can be written as $\frac{a}{b}$ for positive coprime integers $a, b$ . Find $a + b$ .

The answer is 863.

1 solution

Aditya Kumar
Jun 22, 2015

$I=\int _{ 0 }^{ 1 }{ \frac { { \left( 1-16{ x }^{ \frac { 1 }{ 4 } } \right) }^{ 2 } }{ { \left( { 2 }^{ 8 }x \right) }^{ \frac { 7 }{ 8 } } } \, dx } \\ Let\quad x\quad =\quad { t }^{ 8 }\\ \Rightarrow \quad dx\quad =\quad { 8t }^{ 7 }dt\\ \therefore \quad I=\frac { 1 }{ { 2 }^{ 7 } } \int _{ 0 }^{ 1 }{ \frac { { (1-{ 16t }^{ 2 }) }^{ 2 }*{ 8t }^{ 7 }dt }{ { t }^{ 7 } } } \\ \quad \quad \quad =\frac { 1 }{ 16 } \int _{ 0 }^{ 1 }{ \frac { { (1-{ 16t }^{ 2 }) }^{ 2 }dt }{ 1 } } \\ \quad \quad \quad =\frac { 623 }{ 240 }$

Small looking integrals have really big solutions

The answer is 863.

1 solution

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