My problems 16

Calculus Level 4

∫ 0 1 x 6 1 − x 2 d x \large \int_0^1 x^6 \sqrt{1-x^2} \, dx

If the integral above can be expressed as a π b \frac {a\pi}b for coprime positive integers a a and b b , what is the value of a + b a+b ?


The answer is 261.

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Siddharth Bhatt by siddharth bhatt , 25, India

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2 solutions

Aman Rajput
Jun 19, 2015

Put x 2 = t x^2 = t

= ∫ 0 1 t 5 / 2 ( 1 − t ) 1 / 2 d t \displaystyle\int\limits_0^1 t^{5/2}(1-t)^{1/2} dt

= Γ ( 5 / 2 + 1 ) Γ ( 1 / 2 + 1 ) Γ ( 5 / 2 + 1 + 7 / 2 + 1 ) \displaystyle\frac{\Gamma(5/2 + 1)\Gamma(1/2 + 1)}{\Gamma(5/2 + 1 + 7/2 + 1)}

= 5 π 256 \frac{5\pi}{256}

6 Helpful 0 Interesting 0 Brilliant 0 Confused

Even I did it through the Beta Function (first substituting x = sin ⁡ ( y ) x=\sin(y) . However, these approaches would all need a Calculator to evaluate the Beta or Gamma Function. Is there a way you can think of which would not use the Beta or Gamma Function?

User 123 - 5 years, 11 months ago

Beta function works well! +1.

Aditya Kumar - 5 years, 11 months ago
Rushikesh Joshi
Mar 16, 2015

Put x=sint. dx=costdt. Apply Wallis formula.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

thanks for this new approach

Shashank Rustagi - 5 years, 12 months ago

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