find ∫_0 ^1 arctan(x^5 )dx


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Question Number 206224 by mathzup last updated on 09/Apr/24

$f i n d \int_{0}^{1} a r c t a n (x^{5}) d x$
	Answered by Frix last updated on 10/Apr/24

	$\int \tan^{- 1} x^{n} d x \underset{u^{'} = 1 \land v = \tan^{- 1} x^{n}}{\overset{by parts}{=}}$ $= x \tan^{- 1} x^{n} - n \int \frac{x^{n}}{x^{2 n} + 1} d x$ $With n = 5 and x \in [0, 1] we get$ $\frac{\sqrt{5} \ln (1 + \sqrt{5})}{2} - \frac{(1 + \sqrt{5}) \ln 2}{2} + \frac{5 - \sqrt{10 (5 - \sqrt{5})}}{20} π \approx . 151019250$
	Answered by mathzup last updated on 10/Apr/24

	$t h a n k y o u s i r$
	Answered by mathzup last updated on 10/Apr/24
	$by parts we get I=[x arctan(x^5 )]_0 ^1 −∫_0 ^1 x.((5x^4 )/(1+x^(10) ))dx =(π/4)−5∫_0 ^1 (x^5 /(1+x^(10) ))dx but ∫_0 ^1 (x^5 /(1+x^(10) ))dx=∫_0 ^1 x^5 Σ(−1)^n x^(10n) dx =Σ_(n=0) ^∞ (−1)^n ∫_0 ^1 x^(10n+5) dx =Σ_(n=0) ^∞ (−1)^n ×(1/(10n+6)) =Σ_(n=o) ^∞ (1/(10(2n)+6))−Σ_(n=0) ^∞ (1/(10(2n+1)+6)) =Σ_(n=0) ^∞ ((1/(20n+6))−(1/(20n+16))) =10Σ_(n=0) ^∞ (1/((20n+6)(20n+16))) =((10)/(20^2 ))Σ_(n=0) ^∞ (1/((n+(6/(20)))(n+((16)/(20))))) =(1/(40))×((Ψ(((16)/(20)))−Ψ((6/(20))))/(((16)/(20))−(6/(20))))=(1/(20)){Ψ((4/5))−Ψ((3/(10)))}$
	$b y p a r t s w e g e t$ $I = {[x a r c t a n (x^{5})]}_{0}^{1} - \int_{0}^{1} x . \frac{5 x^{4}}{1 + x^{10}} d x$ $= \frac{π}{4} - 5 \int_{0}^{1} \frac{x^{5}}{1 + x^{10}} d x b u t$ $\int_{0}^{1} \frac{x^{5}}{1 + x^{10}} d x = \int_{0}^{1} x^{5} Σ {(- 1)}^{n} x^{10 n} d x$ $= \sum_{n = 0}^{\infty} {(- 1)}^{n} \int_{0}^{1} x^{10 n + 5} d x$ $= \sum_{n = 0}^{\infty} {(- 1)}^{n} \times \frac{1}{10 n + 6}$ $= \sum_{n = o}^{\infty} \frac{1}{10 (2 n) + 6} - \sum_{n = 0}^{\infty} \frac{1}{10 (2 n + 1) + 6}$ $= \sum_{n = 0}^{\infty} (\frac{1}{20 n + 6} - \frac{1}{20 n + 16})$ $= 10 \sum_{n = 0}^{\infty} \frac{1}{(20 n + 6) (20 n + 16)}$ $= \frac{10}{20^{2}} \sum_{n = 0}^{\infty} \frac{1}{(n + \frac{6}{20}) (n + \frac{16}{20})}$ $= \frac{1}{40} \times \frac{Ψ (\frac{16}{20}) - Ψ (\frac{6}{20})}{\frac{16}{20} - \frac{6}{20}} = \frac{1}{20} {Ψ (\frac{4}{5}) - Ψ (\frac{3}{10})}$
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