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x^x =e^(xln x) e^x =1+x+(x^2 /(2!))+(x^3 /(3!))+... ∴ e^(xln x) = 1+xln(x)+(((xln x)^2 )/(2!))+... e^(xln x) = (((xln x)^0 )/(0!))+(((xln x)^1 )/(1!))+(((xln x)^2 )/(2!))+... x^x =e^(xln x) =Σ_(n=0) ^∞ ((x^n (ln x)^n )/(n!)) Question: ∫_0 ^( 1) x^x dx=Σ_(i=0) ^∞ ∫_0 ^1 ((x^n (ln x)^n )/(n!))dx by substituting: x=exp(−(u/(n+1))), 0<u<∞ show that: ∫_0 ^( 1) x^n (ln x)^n =(−1)^n (n+1)^(−(n+1)) ∫_0 ^( ∞) u^n e^(−u) du

$x^{x} = e^{x \ln x}$ $e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + . . .$ $∴ e^{x \ln x} = 1 + x \ln (x) + \frac{{(x \ln x)}^{2}}{2!} + . . .$ $e^{x \ln x} = \frac{{(x \ln x)}^{0}}{0!} + \frac{{(x \ln x)}^{1}}{1!} + \frac{{(x \ln x)}^{2}}{2!} + . . .$ $x^{x} = e^{x \ln x} = \underset{n = 0}{\sum^{\infty}} \frac{x^{n} {(\ln x)}^{n}}{n!}$ $Question :$ $\int_{0}^{1} x^{x} d x = \underset{i = 0}{\sum^{\infty}} \int_{0}^{1} \frac{x^{n} {(\ln x)}^{n}}{n!} d x$ $by substituting :$ $x = \exp (- \frac{u}{n + 1}), 0 < u < \infty$ $show that :$ $\int_{0}^{1} x^{n} {(\ln x)}^{n} = {(- 1)}^{n} {(n + 1)}^{- (n + 1)} \int_{0}^{\infty} u^{n} e^{- u} d u$

Question Number 6021 Answers: 1 Comments: 3

2^x + 2x = 8 find the value of x

$2^{x} + 2 x = 8$ $f i n d t h e v a l u e o f x$

Question Number 6020 Answers: 0 Comments: 0

Question Number 6016 Answers: 1 Comments: 0

∫(((2x + 5)/(√(3 + 4x − 5x^2 ))))dx please help.

$\int (\frac{2 x + 5}{\sqrt{3 + 4 x - 5 x^{2}}}) d x$ $p l e a s e h e l p .$

Question Number 6011 Answers: 0 Comments: 0

Question Number 6009 Answers: 0 Comments: 0

Is there a general solution to: S=Σ_(i=1) ^n i^k , k∈Z, k≥1

$Is there a general solution to :$ $S = \underset{i = 1}{\sum^{n}} i^{k}, k \in Z, k ⩾ 1$

Question Number 5998 Answers: 1 Comments: 5

determine equation of circle that offensive the both of coordinate and through (2,−1)

$d e t e r m i n e e q u a t i o n o f c i r c l e t h a t o f f e n s i v e t h e$ $b o t h o f c o o r d i n a t e a n d t h r o u g h (2, - 1)$

Question Number 5991 Answers: 1 Comments: 0

Solve the differential equation cos^2 (x) (dy/dx) + y = tan(x)

$S o l v e t h e d i f f e r e n t i a l e q u a t i o n$ ${c o s}^{2} (x) \frac{d y}{d x} + y = t a n (x)$

Question Number 5989 Answers: 1 Comments: 0

Evaluate 10 ×12 × 14 × 16 × 18 × 20 into factorial form

$E v a l u a t e 10 \times 12 \times 14 \times 16 \times 18 \times 20 i n t o f a c t o r i a l f o r m$

Question Number 5988 Answers: 1 Comments: 0

If ur = log r show that Σ_(r = 1) ^(10) ur = log3628800 please help.

$I f u r = l o g r$ $s h o w t h a t \sum_{r = 1}^{10} u r = l o g 3628800$ $p l e a s e h e l p .$

Question Number 6076 Answers: 0 Comments: 4

Question about Geogebra •I have drawn a circle in Geogebra. Now I want to fill it with colour (because I need a circular region). Is there a way? •Is there a way to fill with colour any closed figure composed of line-segment/s and arc/s?

$Q u e s t i o n a b o u t G e o g e b r a$ $∙ I h a v e d r a w n a c i r c l e i n$ $G e o g e b r a . N o w I w a n t t o$ $f i l l i t w i t h c o l o u r (b e c a u s e$ $I n e e d a c i r c u l a r r e g i o n) .$ $I s t h e r e a w a y ?$ $∙ I s t h e r e a w a y t o f i l l w i t h$ $c o l o u r a n y c l o s e d f i g u r e$ $c o m p o s e d o f l i n e - s e g m e n t / s$ $a n d a r c / s ?$

Question Number 6079 Answers: 0 Comments: 0

∫((x+cos 2x)/(x+sin 2x))dx=?

$\int \frac{x + \cos 2 x}{x + \sin 2 x} d x = ?$

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