Question and Answers Forum

IntegrationQuestion and Answers: Page 226

Question Number 63667 Answers: 0 Comments: 3

1) calculate ∫_0 ^(2π) (dt/(cost +x sint)) wih x from R. 2) calculate ∫_0 ^(2π) ((sint)/((cost +xsint)^2 ))dt 3) find[the value of ∫_0 ^(2π) (dt/(cos(2t)+2sin(2t)))

$1) c a l c u l a t e \int_{0}^{2 π} \frac{d t}{c o s t + x s i n t} w i h x f r o m R .$ $2) c a l c u l a t e \int_{0}^{2 π} \frac{s i n t}{{(c o s t + x s i n t)}^{2}} d t$ $3) f i n d [t h e v a l u e o f \int_{0}^{2 π} \frac{d t}{c o s (2 t) + 2 s i n (2 t)}$

Question Number 63666 Answers: 0 Comments: 3

calculate ∫_0 ^(2π) (dx/(2sinx +cosx))

$c a l c u l a t e \int_{0}^{2 π} \frac{d x}{2 s i n x + c o s x}$

Question Number 63664 Answers: 0 Comments: 6

let f(x)=∫_0 ^∞ (t^(a−1) /(x+t^n )) dt with 0<a<1 and x>0 and n≥2 1) determine a explicit form of f(x) 2) calculate g(x) =∫_0 ^∞ (t^(a−1) /((x+t^n )^2 )) dt 3) find f^((k)) (x) at form of integrals 4) calculate ∫_0 ^∞ (t^(a−1) /(9+t^2 )) dt and ∫_0 ^∞ (t^(a−1) /((9+t^2 )^2 )) 5) calculate U_n =∫_0 ^∞ (t^((1/n)−1) /(2^n +t^n )) dt and study the convergence of Σ U_n

$l e t f (x) = \int_{0}^{\infty} \frac{t^{a - 1}}{x + t^{n}} d t w i t h 0 < a < 1 a n d x > 0 a n d n ⩾ 2$ $1) d e t e r m i n e a e x p l i c i t f o r m o f f (x)$ $2) c a l c u l a t e g (x) = \int_{0}^{\infty} \frac{t^{a - 1}}{{(x + t^{n})}^{2}} d t$ $3) f i n d f^{(k)} (x) a t f o r m o f i n t e g r a l s$ $4) c a l c u l a t e \int_{0}^{\infty} \frac{t^{a - 1}}{9 + t^{2}} d t a n d \int_{0}^{\infty} \frac{t^{a - 1}}{{(9 + t^{2})}^{2}}$ $5) c a l c u l a t e U_{n} = \int_{0}^{\infty} \frac{t^{\frac{1}{n} - 1}}{2^{n} + t^{n}} d t a n d s t u d y t h e c o n v e r g e n c e o f Σ U_{n}$

Question Number 63662 Answers: 0 Comments: 1

let A_n =∫_0 ^∞ (x^(a−1) /(1+x^n ))dx with n integr and n≥2 and 0<a<1 1) calculate A_n 2) find the values of ∫_0 ^∞ (x^(a−1) /(1+x^2 ))dx and ∫_0 ^∞ (x^(a−1) /(1+x^3 ))dx 3)calculate ∫_0 ^∞ (dx/((√x)(1+x^4 ))) and ∫_0 ^∞ (dx/((^3 (√x^2 ))(1+x^4 )))

$l e t A_{n} = \int_{0}^{\infty} \frac{x^{a - 1}}{1 + x^{n}} d x w i t h n i n t e g r a n d n ⩾ 2 a n d 0 < a < 1$ $1) c a l c u l a t e A_{n}$ $2) f i n d t h e v a l u e s o f \int_{0}^{\infty} \frac{x^{a - 1}}{1 + x^{2}} d x a n d \int_{0}^{\infty} \frac{x^{a - 1}}{1 + x^{3}} d x$ $3) c a l c u l a t e \int_{0}^{\infty} \frac{d x}{\sqrt{x} (1 + x^{4})} a n d \int_{0}^{\infty} \frac{d x}{(^{3} \sqrt{x^{2}}) (1 + x^{4})}$

Question Number 63661 Answers: 0 Comments: 1

let 0<a<1 find the valueof ∫_0 ^∞ (t^(a−1) /(1+t^2 ))dt

$l e t 0 < a < 1 f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{t^{a - 1}}{1 + t^{2}} d t$

Question Number 63641 Answers: 0 Comments: 2

Question Number 63615 Answers: 0 Comments: 5

Question Number 63570 Answers: 1 Comments: 3

Question Number 63566 Answers: 0 Comments: 2

prove that ∫sin^n (x) dx , p∈n , p≥2 =− (1/n)cos(x) sin^(n−1) (x) + (p−1)∫sin^(n−2) (x) dx

$p r o v e t h a t$ $\int {s i n}^{n} (x) d x, p \in n, p ⩾ 2 = - \frac{1}{n} c o s (x) {s i n}^{n - 1} (x) + (p - 1) \int {s i n}^{n - 2} (x) d x$

Question Number 63519 Answers: 0 Comments: 4

consider the general definite intergral I_n =∫_0 ^(π/2) sin^n xdx a) prove that for n≥2, nI_n =(n−1)I_(n−2) . b) Find the values of i)∫_0 ^(π/2) sin^5 dx ii) ∫_0 ^(π/2) sin^6 dx

$c o n s i d e r t h e g e n e r a l d e f i n i t e i n t e r g r a l$ $I_{n} = \int_{0}^{\frac{π}{2}} {s i n}^{n} x d x$ $a) p r o v e t h a t f o r n ⩾ 2, {n I}_{n} = (n - 1) I_{n - 2} .$ $b) F i n d t h e v a l u e s o f i) \int_{0}^{\frac{π}{2}} {s i n}^{5} d x i i) \int_{0}^{\frac{π}{2}} {s i n}^{6} d x$

Question Number 63510 Answers: 0 Comments: 1

let f(x)=∫_0 ^∞ (t^(a−1) /(x+t)) dt with x>0 and 0<a<1 1)calculate f(x) 2)calculate g(x)=∫_0 ^∞ (t^(a−1) /((x+t)^2 ))dt 3)find the value of∫_0 ^∞ (t^(a−1) /((1+t)^2 ))dt

$l e t f (x) = \int_{0}^{\infty} \frac{t^{a - 1}}{x + t} d t w i t h x > 0$ $a n d 0 < a < 1$ $1) c a l c u l a t e f (x)$ $2) c a l c u l a t e g (x) = \int_{0}^{\infty} \frac{t^{a - 1}}{{(x + t)}^{2}} d t$ $3) f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{t^{a - 1}}{{(1 + t)}^{2}} d t$

Question Number 63509 Answers: 1 Comments: 1

calculate ∫_(−1) ^1 ((√(1+x^2 )) −(√(1−x^2 )))dx

$c a l c u l a t e \int_{- 1}^{1} (\sqrt{1 + x^{2}} - \sqrt{1 - x^{2}}) d x$

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[

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

Question Number 63251 Answers: 0 Comments: 0

∫_( 0) ^( (π/2)) sin^(−1) (m cosθ) dθ

$\int_{0}^{\frac{π}{2}} \sin^{- 1} (m \cos θ) d θ$

Question Number 63214 Answers: 0 Comments: 1

calculate ∫_0 ^∞ x e^(−(x^2 /a^2 )) sin(bx)dx with a>0 and b>0

$c a l c u l a t e \int_{0}^{\infty} x e^{- \frac{x^{2}}{a^{2}}} s i n (b x) d x w i t h a > 0 a n d b > 0$

Question Number 63268 Answers: 0 Comments: 0

Pg 221 Pg 222 Pg 223 Pg 224 Pg 225 Pg 226 Pg 227 Pg 228 Pg 229 Pg 230

Contact: info@tinkutara.com