Definite Integral from 0 to 1 of x^4 (1 - x)^4 over (1 + x^2)
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Theorem
\(\ds \int_0^1 \dfrac {x^4 \paren {1 - x}^4} {1 + x^2} \rd x\) | \(=\) | \(\ds \dfrac {22} 7 - \pi\) | ||||||||||||
\(\ds \) | \(\approx\) | \(\ds 0 \cdotp 001264489\) |
This sequence is A003077 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
\(\ds \int_0^1 \dfrac {x^4 \paren {1 - x}^4} {1 + x^2} \rd x\) | \(=\) | \(\ds \int_0^1 \dfrac {x^4 \paren {1 - 4 x + 6 x^2 - 4 x^3 + x^4} } {1 + x^2} \rd x\) | Fourth Power of Difference | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 \dfrac {x^4} {1 + x^2} \rd x - 4 \int_0^1 \dfrac {x^5} {1 + x^2} \rd x + 6 \int_0^1 \dfrac {x^6} {1 + x^2} \rd x - 4 \int_0^1 \dfrac {x^7} {1 + x^2} \rd x + \int_0^1 \dfrac {x^8} {1 + x^2} \rd x\) | Linear Combination of Definite Integrals |
We then establish a reduction formula:
\(\ds \dfrac {x^m} {1 + x^2}\) | \(=\) | \(\ds \dfrac {x^{m - 2} \paren {1 + x^2} - x^{m - 2} } {1 + x^2}\) | ||||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds x^{m - 2} - \dfrac {x^{m - 2} } {1 + x^2}\) |
Then it is a matter of evaluating the individual integrals.
\(\ds \int_0^1 \dfrac {x^4} {1 + x^2} \rd x\) | \(=\) | \(\ds \int_0^1 \paren {x^2 - \dfrac {x^2} {1 + x^2} } \rd x\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 x^2 \rd x - \int_0^1 \dfrac {x^2} {1 + x^2} \rd x\) | Linear Combination of Definite Integrals | |||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {\dfrac {x^3} 3} 0 1 - \int_0^1 \dfrac {x^2} {1 + x^2} \rd x\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {\dfrac {x^3} 3} 0 1 - \bigintlimits {x - \arctan x} 0 1\) | Primitive of $\dfrac {x^2} {x^2 + a^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 3 - \paren {1 - \arctan 1} - \paren {0 - \arctan 0}\) | plugging in the limits | |||||||||||
\(\text {(2)}: \quad\) | \(\ds \) | \(=\) | \(\ds \dfrac \pi 4 - \dfrac 2 3\) | simplifying |
\(\ds \int_0^1 \dfrac {x^5} {1 + x^2} \rd x\) | \(=\) | \(\ds \int_0^1 \paren {x^3 - \dfrac {x^3} {1 + x^2} } \rd x\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 x^3 \rd x - \int_0^1 \dfrac {x^3} {1 + x^2} \rd x\) | Linear Combination of Definite Integrals | |||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {\dfrac {x^4} 4} 0 1 - \int_0^1 \dfrac {x^3} {1 + x^2} \rd x\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {\dfrac {x^4} 4} 0 1 - \intlimits {\frac {x^2} 2 - \frac {a^2} 2 \, \map \ln {x^2 + a^2} } 0 1\) | Primitive of $\dfrac {x^3} {x^2 + a^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 4 - \paren {\paren {\dfrac 1 2 - 0} - \paren {\frac 1 2 \, \map \ln {1 + 1^2} - \frac 1 2 \, \map \ln {1 + 0^2} } }\) | plugging in the limits | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 4 - \dfrac 1 2 + \frac 1 2 \ln 2\) | simplifying | |||||||||||
\(\text {(3)}: \quad\) | \(\ds \) | \(=\) | \(\ds \dfrac {\ln 2} 2 - \dfrac 1 4\) | simplifying | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds -4 \int_0^1 \dfrac {x^5} {1 + x^2} \rd x\) | \(=\) | \(\ds 1 - 2 \ln 2\) |
\(\ds \int_0^1 \dfrac {x^6} {1 + x^2} \rd x\) | \(=\) | \(\ds \int_0^1 \paren {x^4 - \dfrac {x^4} {1 + x^2} } \rd x\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 x^4 \rd x - \int_0^1 \dfrac {x^4} {1 + x^2} \rd x\) | Linear Combination of Definite Integrals | |||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {\dfrac {x^5} 5} 0 1 - \int_0^1 \dfrac {x^4} {1 + x^2} \rd x\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {\dfrac {x^5} 5} 0 1 - \paren {\dfrac \pi 4 - \dfrac 2 3}\) | from $(2)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 5 - \dfrac \pi 4 + \dfrac 2 3\) | plugging in the limits | |||||||||||
\(\text {(4)}: \quad\) | \(\ds \) | \(=\) | \(\ds \dfrac {13} {15} - \dfrac \pi 4\) | simplifying | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 6 \int_0^1 \dfrac {x^6} {1 + x^2} \rd x\) | \(=\) | \(\ds \dfrac {26} 5 - \dfrac {3 \pi} 2\) |
\(\ds \int_0^1 \dfrac {x^7} {1 + x^2} \rd x\) | \(=\) | \(\ds \int_0^1 \paren {x^5 - \dfrac {x^5} {1 + x^2} } \rd x\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 x^5 \rd x - \int_0^1 \dfrac {x^5} {1 + x^2} \rd x\) | Linear Combination of Definite Integrals | |||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {\dfrac {x^6} 6} 0 1 - \int_0^1 \dfrac {x^5} {1 + x^2} \rd x\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {\dfrac {x^6} 6} 0 1 - \paren {\dfrac {\ln 2} 2 - \dfrac 1 4}\) | from $(3)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 6 - \dfrac {\ln 2} 2 + \dfrac 1 4\) | plugging in the limits | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 5 {12} - \dfrac {\ln 2} 2\) | simplifying | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds -4 \int_0^1 \dfrac {x^7} {1 + x^2} \rd x\) | \(=\) | \(\ds 2 \ln 2 - \dfrac 5 3\) |
\(\ds \int_0^1 \dfrac {x^8} {1 + x^2} \rd x\) | \(=\) | \(\ds \int_0^1 \paren {x^6 - \dfrac {x^6} {1 + x^2} } \rd x\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^1 x^6 \rd x - \int_0^1 \dfrac {x^6} {1 + x^2} \rd x\) | Linear Combination of Definite Integrals | |||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {\dfrac {x^7} 7} 0 1 - \int_0^1 \dfrac {x^6} {1 + x^2} \rd x\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {\dfrac {x^7} 7} 0 1 - \paren {\dfrac {13} {15} - \dfrac \pi 4}\) | from $(4)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 7 - \dfrac {13} {15} + \dfrac \pi 4\) | plugging in the limits |
It remains to gather up the terms.
First the terms in $x^5$ and $x^7$, as there is some important cancelling out:
\(\ds \paren {-4 \int_0^1 \dfrac {x^5} {1 + x^2} \rd x} + \paren {-4 \int_0^1 \dfrac {x^7} {1 + x^2} \rd x}\) | \(=\) | \(\ds \paren {1 - 2 \ln 2} + \paren {2 \ln 2 - \dfrac 5 3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 - \dfrac 5 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac 2 3\) |
Thus we continue:
\(\ds \int_0^1 \dfrac {x^4 \paren {1 - x}^4} {1 + x^2} \rd x\) | \(=\) | \(\ds \paren {\dfrac \pi 4 - \dfrac 2 3} - \dfrac 2 3 + \paren {\dfrac {26} 5 - \dfrac {3 \pi} 2} + \paren {\dfrac 1 7 - \dfrac {13} {15} + \dfrac \pi 4}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-\dfrac 2 3 - \dfrac 2 3} + \dfrac 1 7 + \dfrac {26} 5 - \dfrac {13} {15} - \pi\) | rearranging, and resolving the multiples of $\pi$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 7 + \dfrac {78 - 13 - 10 - 10} {15} - \pi\) | gathering all the multiples of $\dfrac 1 {15}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 3 + \dfrac 1 7 - \pi\) | arithmetic |
The result follows.
$\blacksquare$
Sources
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,001264489 \ldots$