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IntegrationQuestion and Answers: Page 300

Question Number 32338 Answers: 0 Comments: 0

find the value of ∫_0 ^1 ((ln(t))/((1+t)(√(1−t^2 )))) dt.

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

Question Number 32337 Answers: 0 Comments: 0

1)calculate ∫_a ^(+∞) (dx/((1+x^2 )(√(x^2 −a^2 )))) with a>0 2) find the value of ∫_2 ^(+∞) (dx/((1+x^2 )(√(x^2 −4)))) .

$1) c a l c u l a t e \int_{a}^{+ \infty} \frac{d x}{(1 + x^{2}) \sqrt{x^{2} - a^{2}}} w i t h a > 0$ $2) f i n d t h e v a l u e o f \int_{2}^{+ \infty} \frac{d x}{(1 + x^{2}) \sqrt{x^{2} - 4}} .$

Question Number 32323 Answers: 1 Comments: 0

Given f(x) = (3/(16))(∫_0 ^1 f(x)dx)x^2 − (9/(10))(∫_0 ^2 f(x)dx)x + 2(∫_0 ^3 f(x)dx) + 4 Find f(x)

$Given$ $f (x) = \frac{3}{16} (\int_{0}^{1} f (x) d x) x^{2} - \frac{9}{10} (\int_{0}^{2} f (x) d x) x + 2 (\int_{0}^{3} f (x) d x) + 4$ $Find f (x)$

Question Number 32304 Answers: 0 Comments: 0

find lim_(x→+∞) e^(−x^2 ) ∫_0 ^x e^t^2 dt .

$f i n d {l i m}_{x \to + \infty} e^{- x^{2}} \int_{0}^{x} e^{t^{2}} d t .$

Question Number 32302 Answers: 1 Comments: 0

calculate ∫_1 ^2 (dx/(x +x(√x))) .

$c a l c u l a t e \int_{1}^{2} \frac{d x}{x + x \sqrt{x}} .$

Question Number 32301 Answers: 0 Comments: 1

calculate ∫_1 ^e ln(1+(√x))dx .

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

Question Number 32269 Answers: 1 Comments: 0

find ∫ (x^3 /(√(1+x^2 ))) dx

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

Question Number 32258 Answers: 2 Comments: 0

find ∫ (1/(2−x^2 )) dx

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

Question Number 32206 Answers: 0 Comments: 0

Find Σ_(k=1) ^∞ (∫_(k−1) ^k x^(−x) dx) .

$Find \underset{k = 1}{\sum^{\infty}} (\underset{k - 1}{\int^{k}} x^{- x} dx) .$

Question Number 32139 Answers: 0 Comments: 4

Find the ∫ ((x+1)/(x^2 +x+1))dx

$F i n d t h e$ $\int \frac{x + 1}{x^{2} + x + 1} d x$

Question Number 32045 Answers: 0 Comments: 0

find lim_(n→∞) ∫_0 ^∞ e^(−t) sin^n t dt .

$f i n d {l i m}_{n \to \infty} \int_{0}^{\infty} e^{- t} {s i n}^{n} t d t .$

Question Number 32044 Answers: 0 Comments: 1

fimd lim_(x→0) (1/x^3 ) ∫_0 ^x t^2 ln(1+sint) dt .

$f i m d {l i m}_{x \to 0} \frac{1}{x^{3}} \int_{0}^{x} t^{2} l n (1 + s i n t) d t .$

Question Number 32043 Answers: 0 Comments: 0

let f(x)= ∫_x ^x^2 (dt/(lnt)) with x>0 and x≠1 1) prove that ∀ x>1 ∫_x ^x^2 ((xdt)/(tlnt)) ≤f(x)≤ ∫_x ^x^2 ((x^2 dt)/(tlnt)) after find lim_(x→1) f(x) 2) calculate f^′ (x) .

$l e t f (x) = \int_{x}^{x^{2}} \frac{d t}{l n t} w i t h x > 0 a n d x \neq 1$ $1) p r o v e t h a t \forall x > 1 \int_{x}^{x^{2}} \frac{x d t}{t l n t} ⩽ f (x) ⩽ \int_{x}^{x^{2}} \frac{x^{2} d t}{t l n t} a f t e r$ $f i n d {l i m}_{x \to 1} f (x)$ $2) c a l c u l a t e f^{^{'}} (x) .$

Question Number 32040 Answers: 0 Comments: 2

let give f(x) =∫_0 ^(π/2) (dt/(1+x tant)) 1) find a simple form of f(x) 2) calculate ∫_0 ^(π/2) ((tant)/((1+xtant)^2 ))dt 3)give the value of ∫_0 ^(π/2) ((tant)/((1+(√3) tant)^2 )) dt .

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

Question Number 32039 Answers: 0 Comments: 3

a>−1 calculate ∫_0 ^(π/2) (dt/(1+a tan^2 t)) . 2) find ∫_0 ^(π/2) ((tan^2 t)/((1+atan^2 t)^2 )) dt 3) find the value of ∫_0 ^(π/2) ((tan^2 t)/((1+2tan^2 t)^2 ))dt.

$a > - 1 c a l c u l a t e \int_{0}^{\frac{π}{2}} \frac{d t}{1 + a {t a n}^{2} t} .$ $2) f i n d \int_{0}^{\frac{π}{2}} \frac{{t a n}^{2} t}{{(1 + {a t a n}^{2} t)}^{2}} d t$ $3) f i n d t h e v a l u e o f \int_{0}^{\frac{π}{2}} \frac{{t a n}^{2} t}{{(1 + 2 {t a n}^{2} t)}^{2}} d t .$

Question Number 32034 Answers: 0 Comments: 0

let u_n = ∫_0 ^1 (dx/(1+x+...+x^n )) study the convergence of Σ u_n .

$l e t u_{n} = \int_{0}^{1} \frac{d x}{1 + x + . . . + x^{n}} s t u d y t h e c o n v e r g e n c e o f$ $Σ u_{n} .$

Question Number 32031 Answers: 0 Comments: 1

let f(a) = ∫_0 ^∞ e^(−ax) ln(x)dx with a>0 1) find f(a) 2) find ∫_0 ^∞ e^(−ax) (xlnx)dx 3) calculate ∫_0 ^∞ e^(−2x) (xlnx)dx .

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

Question Number 32029 Answers: 0 Comments: 0

calculate ∫_0 ^∞ e^(−αx) ln(x) dx with α>0 .

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

Question Number 32026 Answers: 0 Comments: 1

let α>0 prove that Σ_(n=0) ^∞ (((−1)^n )/(n+α)) =∫_0 ^1 (x^(α−1) /(1+x))dx .

$l e t α > 0 p r o v e t h a t \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n + α} = \int_{0}^{1} \frac{x^{α - 1}}{1 + x} d x .$

Question Number 31974 Answers: 0 Comments: 0

1)find I(p,q) = ∫_0 ^1 t^p (1−t)^q dt with pand q integrs 2) find the nature of Σ I_((n,n))

$1) f i n d I (p, q) = \int_{0}^{1} t^{p} {(1 - t)}^{q} d t w i t h p a n d q i n t e g r s$ $2) f i n d t h e n a t u r e o f Σ I_{(n, n)}$

Question Number 31970 Answers: 0 Comments: 0

find the nature of ∫_2 ^∞ (e^(−x) /(√(x^2 −4))) dx .

$f i n d t h e n a t u r e o f \int_{2}^{\infty} \frac{e^{- x}}{\sqrt{x^{2} - 4}} d x .$

Question Number 31969 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ (((1+t^2 )/(1+t^4 )))arctant dt.

$f i n d t h e v a l u e o f \int_{0}^{\infty} (\frac{1 + t^{2}}{1 + t^{4}}) a r c t a n t d t .$

Question Number 31968 Answers: 0 Comments: 0

find ∫_2 ^(√5) x(√((x−2)((√5)−x))) dx .

$f i n d \int_{2}^{\sqrt{5}} x \sqrt{(x - 2) (\sqrt{5} - x)} d x .$

Question Number 31967 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ ((arctanx)/(x^2 +x+1))dx .

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

Question Number 31951 Answers: 1 Comments: 0

Evaluate ∫sin (√x)dx

$E v a l u a t e \int \sin \sqrt{x} d x$

Question Number 31858 Answers: 0 Comments: 1

∫((sinx)/x)dx

$\int \frac{s i n x}{x} d x$

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