Question

Evaluate `int_(0)^(1)x^(4)(1-sqrt(x))^(5)dx`​

Answer 1

Answer:

∫101+x4−−−−−−√dx

I used substitution of tanx=z but it was not fruitful. Then i used (x−1/x)=z and (x)2−1/(x)2=z but no helpful expression was derived. I also used property ∫a0f(a−x)=∫a0f(x) Please help me out

calculus

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edited Jun 14 '17 at 0:43

Felix Marin

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asked Jun 6 '17 at 5:01

Sourav

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1

Throw the command Int[Sqrt[1+x^4],{x,0,1}] to WA, the integral evaluates to 2F1(−12,14;−54;−1)≈1.0894294132248223224… where 2F1 is the hypergeometric function. There is probably no elementary way to evaluate the integral. – achille hui Jun 6 '17 at 5:18

Thank you...trying to understant your answer – Sourav Jun 6 '17 at 5:40

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5 Answers

18

We can do better than hypergeometric function and elliptic integral:

∫101+x4−−−−−√dx=2–√3+Γ2(14)12π−−√

Firstly, integration by part gives

∫101+x4−−−−−√dx=2–√−2∫10x41+x4−−−−−√dx=2–√−2∫10(1+x4−−−−−√−11+x4−−−−−√)dx

Hence

∫101+x4−−−−−√dx=2–√3+23∫1011+x4−−−−−√dx

Making x=1/u in the last integral gives

∫1011+x4−−−−−√dx=12∫∞011+x4−−−−−√dx=18π−−√Γ2(14)

Step-by-step explanation: