Question and Answers Forum

let give J(x)= (1/π) ∫_0 ^π cos(xcost)dt 1) find J^′ and J^(′′) in form of integrals 2)prove that J^′ (x)=((−x)/π) ∫_0 ^π sin^2 t cos(xcost)dt and J is solution of d.e. xy^(′′) +y^′ +xy=0

$l e t g i v e J (x) = \frac{1}{π} \int_{0}^{π} c o s (x c o s t) d t$ $1) f i n d J^{^{'}} a n d J^{^{″}} i n f o r m o f i n t e g r a l s$ $2) p r o v e t h a t J^{^{'}} (x) = \frac{- x}{π} \int_{0}^{π} {s i n}^{2} t c o s (x c o s t) d t a n d J i s$ $s o l u t i o n o f d . e . {x y}^{^{″}} + y^{^{'}} + x y = 0$

Question Number 30185 Answers: 0 Comments: 0

let I= ∫_0 ^(π/2) ((sinx)/(√(1+sinxcosx)))dx and J= ∫_0 ^(π/2) ((cosx)/(√(1+sinx cosx))) dx 1) calculate I +J 2) find I and J.

$l e t I = \int_{0}^{\frac{π}{2}} \frac{s i n x}{\sqrt{1 + s i n x c o s x}} d x a n d$ $J = \int_{0}^{\frac{π}{2}} \frac{c o s x}{\sqrt{1 + s i n x c o s x}} d x$ $1) c a l c u l a t e I + J$ $2) f i n d I a n d J .$

Question Number 30184 Answers: 0 Comments: 1

find ∫_(1/2) ^2 (1+(1/x^2 ))arctanx dx . (arctan=tan^(−1) ).

$f i n d \int_{\frac{1}{2}}^{2} (1 + \frac{1}{x^{2}}) a r c t a n x d x . (a r c t a n = {t a n}^{- 1}) .$

Question Number 30182 Answers: 0 Comments: 2

find ∫_2 ^3 ((√(x+1))/(x(√(1−x))))dx .

$f i n d \int_{2}^{3} \frac{\sqrt{x + 1}}{x \sqrt{1 - x}} d x .$

Question Number 30181 Answers: 0 Comments: 0

find ∫ (dx/(1+x^3 +x^6 )) .

$f i n d \int \frac{d x}{1 + x^{3} + x^{6}} .$

Question Number 30180 Answers: 0 Comments: 1

find ∫_0 ^(π/2) ((x sinx cosx)/(tan^2 x +cotan^2 x))dx .(use the ch.x=(π/2) −t).

$f i n d \int_{0}^{\frac{π}{2}} \frac{x s i n x c o s x}{{t a n}^{2} x + {c o t a n}^{2} x} d x . (u s e t h e c h . x = \frac{π}{2} - t) .$

Question Number 30179 Answers: 0 Comments: 1

find ∫ (dt/(1+cost +sint)) .

$f i n d \int \frac{d t}{1 + c o s t + s i n t} .$

Question Number 30178 Answers: 0 Comments: 1

calculate ∫_0 ^(π/2) (dx/(1+cosx cosθ)) with −π<θ<π .

$c a l c u l a t e \int_{0}^{\frac{π}{2}} \frac{d x}{1 + c o s x c o s θ} w i t h - π < θ < π .$

Question Number 30008 Answers: 0 Comments: 1

integrate w.r.t x ∫(e^x^2 )dx

$i n t e g r a t e w . r . t x$ $\int (e^{x^{2}}) d x$

Question Number 29980 Answers: 0 Comments: 0

prove that γ= Σ_(n=1) ^∞ ((1/n) −ln(1 +(1/n))) 2)show that γ= Σ_(k=2) ^∞ (((−1)^k )/k) ξ(k).

$p r o v e t h a t γ = \sum_{n = 1}^{\infty} (\frac{1}{n} - l n (1 + \frac{1}{n}))$ $2) s h o w t h a t γ = \sum_{k = 2}^{\infty} \frac{{(- 1)}^{k}}{k} ξ (k) .$

Question Number 29976 Answers: 0 Comments: 0

prove that ln(Γ(x))= −lnx −γx +Σ_(n=1) ^∞ ( (x/n) −ln( 1+(x/n))) with x>0

$p r o v e t h a t$ $l n (Γ (x)) = - l n x - γ x + \sum_{n = 1}^{\infty} (\frac{x}{n} - l n (1 + \frac{x}{n})) w i t h x > 0$

Question Number 29975 Answers: 0 Comments: 2

let give 0<α<1 1) prove that π coth(πα) −(1/α) = Σ_(n=1) ^∞ ((2α)/(α^2 +n^2 )). 2)by integration on[0,1] find Π_(n=1) ^∞ (1+(1/n^2 )).

$l e t g i v e 0 < α < 1$ $1) p r o v e t h a t π c o t h (π α) - \frac{1}{α} = \sum_{n = 1}^{\infty} \frac{2 α}{α^{2} + n^{2}} .$ $2) b y i n t e g r a t i o n o n [0, 1] f i n d \prod_{n = 1}^{\infty} (1 + \frac{1}{n^{2}}) .$

Question Number 29972 Answers: 0 Comments: 1

let give ∣x∣<1 find ∫_0 ^(π/2) (dθ/(√(1−x^2 cos^2 θ))) .

$l e t g i v e ∣ x ∣< 1 f i n d \int_{0}^{\frac{π}{2}} \frac{d θ}{\sqrt{1 - x^{2} {c o s}^{2} θ}} .$

Question Number 29971 Answers: 0 Comments: 2

find J(x)= ∫_0 ^∞ (dt/(x+e^t )) ?.

$f i n d J (x) = \int_{0}^{\infty} \frac{d t}{x + e^{t}} ? .$

Question Number 29957 Answers: 1 Comments: 0

∫3xdx

$\int 3 x d x$

Question Number 29857 Answers: 0 Comments: 0

find ∫_0 ^(+∞) ((ln(x))/((1+x)^3 ))dx .

$f i n d \int_{0}^{+ \infty} \frac{l n (x)}{{(1 + x)}^{3}} d x .$

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