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IntegrationQuestion and Answers: Page 228

Question Number 63508 Answers: 0 Comments: 4

let f(x) =∫_(−∞) ^(+∞) (dt/((t^2 +ixt −1))) with ∣x∣>2 (i^2 =−1) 1) extract Re(f(x)) and Im(f(x)) 2) calculate f(x) 3) find olso g(x) =∫_(−∞) ^(+∞) (t/((t^2 +ixt −1)^2 ))dt 4) find values of integrals ∫_(−∞) ^(+∞) (dt/((t^2 +3it −1))) and ∫_(−∞) ^(+∞) ((tdt)/((t^2 +3it −1)^2 )) 5) give f^((n)) (x) at form of integrals.

$l e t f (x) = \int_{- \infty}^{+ \infty} \frac{d t}{(t^{2} + i x t - 1)} w i t h ∣ x ∣> 2 (i^{2} = - 1)$ $1) e x t r a c t R e (f (x)) a n d I m (f (x))$ $2) c a l c u l a t e f (x)$ $3) f i n d o l s o g (x) = \int_{- \infty}^{+ \infty} \frac{t}{{(t^{2} + i x t - 1)}^{2}} d t$ $4) f i n d v a l u e s o f i n t e g r a l s \int_{- \infty}^{+ \infty} \frac{d t}{(t^{2} + 3 i t - 1)} a n d \int_{- \infty}^{+ \infty} \frac{t d t}{{(t^{2} + 3 i t - 1)}^{2}}$ $5) g i v e f^{(n)} (x) a t f o r m o f i n t e g r a l s .$

Question Number 63433 Answers: 1 Comments: 1

∫_1 ^x x^2 −3x(√x)dx =((−716)/(15)) then calculate ∫_x ^(x+1) (1/(x+3))dx

$\underset{1}{\int^{x}} x^{2} - 3 x \sqrt{x} d x = \frac{- 716}{15}$ $t h e n c a l c u l a t e \underset{x}{\int^{x + 1}} \frac{1}{x + 3} d x$

Question Number 63410 Answers: 2 Comments: 2

Question Number 63405 Answers: 0 Comments: 0

find ∫ (√((x^2 −4x+1)/(x+2)))dx

$f i n d \int \sqrt{\frac{x^{2} - 4 x + 1}{x + 2}} d x$

Question Number 63404 Answers: 1 Comments: 2

1) find ∫ ((x+1)/(x^3 −3x −2))dx 2) calculate ∫_4 ^(+∞) ((x+1)/(x^3 −3x +2))dx

$1) f i n d \int \frac{x + 1}{x^{3} - 3 x - 2} d x$ $2) c a l c u l a t e \int_{4}^{+ \infty} \frac{x + 1}{x^{3} - 3 x + 2} d x$

Question Number 63395 Answers: 0 Comments: 3

let f(t) =∫_0 ^∞ ((ln(1+tx))/(1+x^2 ))dx with ∣t∣<1 1) determine a explicit form of f(t) 2) find the value of ∫_0 ^∞ ((ln(1+x))/(1+x^2 ))dx

$l e t f (t) = \int_{0}^{\infty} \frac{l n (1 + t x)}{1 + x^{2}} d x w i t h ∣ t ∣< 1$ $1) d e t e r m i n e a e x p l i c i t f o r m o f f (t)$ $2) f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{l n (1 + x)}{1 + x^{2}} d x$

Question Number 63351 Answers: 3 Comments: 2

Question Number 63372 Answers: 1 Comments: 2

For what values of a and b will the integral ∫_a ^b (√(10−x−x^2 ))dx be at maximum

$F o r w h a t v a l u e s o f a a n d b w i l l t h e$ $i n t e g r a l \int_{a}^{b} \sqrt{10 - x - x^{2}} d x b e a t$ $m a x i m u m$

Question Number 63273 Answers: 0 Comments: 1

let F(x) =∫_x^2 ^x^3 ((sin(t))/(t+x)) dt 1) calculate lim_(x→0) F(x) and lim_(x→+∞) F(x) 2)calculste lim_(x→0) F^′ (x) and lim_(x→+∞) F^′ (x)

$l e t F (x) = \int_{x^{2}}^{x^{3}} \frac{s i n (t)}{t + x} d t$ $1) c a l c u l a t e {l i m}_{x \to 0} F (x) a n d {l i m}_{x \to + \infty} F (x)$ $2) c a l c u l s t e {l i m}_{x \to 0} F^{^{'}} (x) a n d {l i m}_{x \to + \infty} F^{^{'}} (x)$

Question Number 63261 Answers: 0 Comments: 6

∫x tan(x) dx

$\int x t a n (x) d x$

Question Number 63232 Answers: 0 Comments: 2

let B(x,y) =∫_0 ^1 (1−t)^(x−1) t^(y−1) dt 1) study the convergence of B(x,y) 1) prove that B(x,y)=B(y,x) prove that B(x,y) =∫_0 ^∞ (t^(x−1) /((1+t)^(x+y) )) dt 2) prove that B(x,y) =((Γ(x).Γ(y))/(Γ(x+y))) 3) prove that Γ(x).Γ(1−x) =(π/(sin(πx))) for allx ∈]0,1[

$l e t B (x, y) = \int_{0}^{1} {(1 - t)}^{x - 1} t^{y - 1} d t$ $1) s t u d y t h e c o n v e r g e n c e o f B (x, y)$ $1) p r o v e t h a t B (x, y) = B (y, x)$ $p r o v e t h a t B (x, y) = \int_{0}^{\infty} \frac{t^{x - 1}}{{(1 + t)}^{x + y}} d t$ $2) p r o v e t h a t B (x, y) = \frac{Γ (x) . Γ (y)}{Γ (x + y)}$ $3) p r o v e t h a t Γ (x) . Γ (1 - x) = \frac{π}{s i n (π x)} f o r a l l x \in] 0, 1 [$