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IntegrationQuestion and Answers: Page 248

Question Number 50659 Answers: 2 Comments: 4

∫_(0 ) ^( ∞) (dx/(4−x^2 )) = ?

$\int_{0}^{\infty} \frac{d x}{4 - x^{2}} = ?$

Question Number 50633 Answers: 1 Comments: 1

please help integrate ((x+sin x)/(1+cos x))dx

$p l e a s e h e l p i n t e g r a t e \frac{x + \sin x}{1 + \cos x} d x$

Question Number 50433 Answers: 0 Comments: 0

find all function f C^2 onR / f(x)+∫_0 ^x (x−t)f(t)dt =1 ∀ x∈R .

$f i n d a l l f u n c t i o n f C^{2} o n R /$ $f (x) + \int_{0}^{x} (x - t) f (t) d t = 1 \forall x \in R .$

Question Number 50427 Answers: 0 Comments: 2

let I_n (λ) =∫_0 ^π ((vos(nt))/(1−2λcost +λ^2 ))dt 1)calculate I_0 (λ) and I_1 (λ) 2) find relation between I_(n−1) ,I_n and I_(n+1) 3) calculate I_n (λ).

$l e t I_{n} (λ) = \int_{0}^{π} \frac{v o s (n t)}{1 - 2 λ c o s t + λ^{2}} d t$ $1) c a l c u l a t e I_{0} (λ) a n d I_{1} (λ)$ $2) f i n d r e l a t i o n b e t w e e n I_{n - 1}, I_{n} a n d I_{n + 1}$ $3) c a l c u l a t e I_{n} (λ) .$

Question Number 50425 Answers: 0 Comments: 0

find ∫_0 ^∞ ((sin^4 (t))/t^3 ) dt

$f i n d \int_{0}^{\infty} \frac{{s i n}^{4} (t)}{t^{3}} d t$

Question Number 50422 Answers: 1 Comments: 1

find ∫_0 ^1 ((ln(x))/((√x)(1−x)^(3/2) ))dx

$f i n d \int_{0}^{1} \frac{l n (x)}{\sqrt{x} {(1 - x)}^{\frac{3}{2}}} d x$

Question Number 50421 Answers: 0 Comments: 1

calculate A =∫_0 ^(π/3) (du/((1+cos^2 u)^3 ))

$c a l c u l a t e A = \int_{0}^{\frac{π}{3}} \frac{d u}{{(1 + {c o s}^{2} u)}^{3}}$

Question Number 50420 Answers: 1 Comments: 4

find ∫_0 ^(π/6) cosx ln(cosx)dx

$f i n d \int_{0}^{\frac{π}{6}} c o s x l n (c o s x) d x$

Question Number 50418 Answers: 0 Comments: 1

calculate ∫_0 ^(lln(3)) ((sh^2 (x)dx)/(ch^3 (x)))

$c a l c u l a t e \int_{0}^{l l n (3)} \frac{{s h}^{2} (x) d x}{{c h}^{3} (x)}$

Question Number 50417 Answers: 0 Comments: 1

find ∫_0 ^1 arctan(√(1−(x^2 /2)))dx

$f i n d \int_{0}^{1} a r c t a n \sqrt{1 - \frac{x^{2}}{2}} d x$

Question Number 50416 Answers: 1 Comments: 0

calculate ∫_0 ^1 ^3 (√(x^2 (1−x^3 )))dx

$c a l c u l a t e \int_{0}^{1}^{3} \sqrt{x^{2} (1 - x^{3})} d x$

Question Number 50415 Answers: 1 Comments: 1

calculate ∫_0 ^(π/2) (dt/(1+cosθ cost))

$c a l c u l a t e \int_{0}^{\frac{π}{2}} \frac{d t}{1 + c o s θ c o s t}$

Question Number 50414 Answers: 1 Comments: 0

calculate ∫_0 ^(π/2) ((x sinx cosx)/(tan^2 x +cotan^2 x))dx ctanx =(1/(tanx))

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

Question Number 50413 Answers: 0 Comments: 1

let f ∈C^0 (R,R) / ∀ x∈R f(a+b−x)=f(x) 1) find ∫_a ^b xf(x)dx interms of ∫_a ^b f(x)dx 2) calculate ∫_0 ^π ((xdx)/(1+sinx))

$l e t f \in C^{0} (R, R) / \forall x \in R f (a + b - x) = f (x)$ $1) f i n d \int_{a}^{b} x f (x) d x i n t e r m s o f \int_{a}^{b} f (x) d x$ $2) c a l c u l a t e \int_{0}^{π} \frac{x d x}{1 + s i n x}$

Question Number 50412 Answers: 0 Comments: 1

1) calculate U_n =∫_0 ^π (dx/(1+cos^2 (nx))) with n from N 2) f continue from [0,π] to R find lim_(n→+∞) ∫_0 ^π ((f(x))/(1+cos^2 (nx)))dx

$1) c a l c u l a t e U_{n} = \int_{0}^{π} \frac{d x}{1 + {c o s}^{2} (n x)} w i t h n f r o m N$ $2) f c o n t i n u e f r o m [0, π] t o R f i n d$ ${l i m}_{n \to + \infty} \int_{0}^{π} \frac{f (x)}{1 + {c o s}^{2} (n x)} d x$

Question Number 50410 Answers: 0 Comments: 0

determine all functions f ∈C^0 (R,R) / ∫_0 ^x f(x)dx =(2/3)xf(x) .

$d e t e r m i n e a l l f u n c t i o n s f \in C^{0} (R, R) /$ $\int_{0}^{x} f (x) d x = \frac{2}{3} x f (x) .$

Question Number 50407 Answers: 0 Comments: 0

determine f ∈C^0 ([0,1],R) verifying ∫_0 ^1 f(x)dx =(1/3) +∫_0 ^1 (f(x^2 ))^2 dx

$d e t e r m i n e f \in C^{0} ([0, 1], R) v e r i f y i n g$ $\int_{0}^{1} f (x) d x = \frac{1}{3} + \int_{0}^{1} {(f (x^{2}))}^{2} d x$

Question Number 50406 Answers: 0 Comments: 2

1) decompose at simple elements U_n = ((n x^(n−1) )/(x^n −1)) 2) calculste ∫_0 ^(2π) (dt/(x−e^(it) ))

$1) d e c o m p o s e a t s i m p l e e l e m e n t s$ $U_{n} = \frac{n x^{n - 1}}{x^{n} - 1}$ $2) c a l c u l s t e \int_{0}^{2 π} \frac{d t}{x - e^{i t}}$

Question Number 50388 Answers: 0 Comments: 0

find inf_((a,b)∈R^2 ) ∫_0 ^1 x^2 (ln(x)−ax−b)^2 dx

$f i n d {i n f}_{(a, b) \in R^{2}} \int_{0}^{1} x^{2} {(l n (x) - a x - b)}^{2} d x$

Question Number 50384 Answers: 1 Comments: 1

find ∫ (dx/((1−x^2 )(1−x^3 ))) 2) calculate ∫_2 ^(√5) (dx/((1−x^2 )(1−x^3 )))

$f i n d \int \frac{d x}{(1 - x^{2}) (1 - x^{3})}$ $2) c a l c u l a t e \int_{2}^{\sqrt{5}} \frac{d x}{(1 - x^{2}) (1 - x^{3})}$

Question Number 50219 Answers: 7 Comments: 0

Question Number 50423 Answers: 1 Comments: 1

find f(a) =∫_a ^(+∞) (dx/((1+x^2 )(√(x^2 −a^2 )))) with a>0

$f i n d f (a) = \int_{a}^{+ \infty} \frac{d x}{(1 + x^{2}) \sqrt{x^{2} - a^{2}}} w i t h a > 0$

Question Number 50186 Answers: 1 Comments: 0

Let f(x)= ∫_2 ^( x) (dt/(1+t^6 )). Prove that : (1/(730))<f(3)<(1/(65)).

$L e t f (x) = \int_{2}^{x} \frac{d t}{1 + t^{6}} .$ $P r o v e t h a t : \frac{1}{730} < f (3) < \frac{1}{65} .$

Question Number 50161 Answers: 1 Comments: 0

Find the function whose first derivative is 8−(5/(x^2 )^(1/3) ) the initial conditions f(8)=−20

$F i n d t h e f u n c t i o n w h o s e f i r s t$ $d e r i v a t i v e i s 8 - \frac{5}{\sqrt[3]{x^{2}}} t h e i n i t i a l$ $c o n d i t i o n s f (8) = - 20$

Question Number 50158 Answers: 7 Comments: 1

Question Number 50052 Answers: 5 Comments: 1

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