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IntegrationQuestion and Answers: Page 282

Question Number 36936 Answers: 0 Comments: 2

calculate I_n = ∫_0 ^π (dx/(1+cos^2 (nx)))

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

Question Number 36935 Answers: 0 Comments: 0

find all function f R→R wich verify ∀(x,y)∈ R^2 f(x).f(y) =∫_(x−y) ^(x+y) f(t)dt .

$f i n d a l l f u n c t i o n f R \to R w i c h v e r i f y$ $\forall (x, y) \in R^{2} f (x) . f (y) = \int_{x - y}^{x + y} f (t) d t .$

Question Number 36932 Answers: 0 Comments: 0

let f ∈ C^0 ([0,π],R) prove that lim_(n→+∞) ∫_0 ^π f(x) ∣sin(nx)∣dx =(2/π) ∫_0 ^π f(x)dx .

$l e t f \in C^{0} ([0, π], R) p r o v e t h a t$ ${l i m}_{n \to + \infty} \int_{0}^{π} f (x) ∣ s i n (n x) ∣ d x = \frac{2}{π} \int_{0}^{π} f (x) d x .$

Question Number 36931 Answers: 0 Comments: 2

calculate ∫_0 ^(2π) (dt/(x −e^(it) ))

$c a l c u l a t e \int_{0}^{2 π} \frac{d t}{x - e^{i t}}$

Question Number 36930 Answers: 0 Comments: 0

let u_n = (1/(2n+1)) +(1/(2n+3)) +.....+(1/(4n−1)) calculate lim_(n→+∞) u_n .

$l e t u_{n} = \frac{1}{2 n + 1} + \frac{1}{2 n + 3} + . . . . . + \frac{1}{4 n - 1}$ $c a l c u l a t e {l i m}_{n \to + \infty} u_{n} .$

Question Number 36919 Answers: 0 Comments: 1

calculate f(α)= ∫_(−∞) ^(+∞) (1+αi)^(−x^2 ) dx .

$c a l c u l a t e f (α) = \int_{- \infty}^{+ \infty} {(1 + α i)}^{- x^{2}} d x .$

Question Number 36918 Answers: 0 Comments: 0

calculate ∫_0 ^(+∞) (1−i)^(−x^2 ) dx

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

Question Number 36917 Answers: 0 Comments: 1

calculate ∫_0 ^(+∞) (1+i)^(−x^2 ) dx

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

Question Number 36916 Answers: 0 Comments: 1

let z=r e^(iθ) fins f(z) = ∫_(−∞) ^(+∞) z^(−x^2 ) dx

$l e t z = r e^{i θ} f i n s f (z) = \int_{- \infty}^{+ \infty} z^{- x^{2}} d x$

Question Number 36915 Answers: 0 Comments: 1

let z =a+ib find f(z) = ∫_(−∞) ^(+∞) z^(−x^2 ) dx

$l e t z = a + i b f i n d f (z) = \int_{- \infty}^{+ \infty} z^{- x^{2}} d x$

Question Number 36912 Answers: 0 Comments: 1

let ⟨p,q⟩= ∫_(−1) ^1 p(x)q(x)dx with p and q are two polynoms fromR[x] 1)let p(x)=x^n calculate ⟨p,p⟩ 2)let p(x)=1+x+x^2 +....+x^n find ⟨p,p⟩.

$l e t ⟨ p, q ⟩ = \int_{- 1}^{1} p (x) q (x) d x w i t h p a n d q a r e$ $t w o p o l y n o m s f r o m R [x]$ $1) l e t p (x) = x^{n} c a l c u l a t e ⟨ p, p ⟩$ $2) l e t p (x) = 1 + x + x^{2} + . . . . + x^{n}$ $f i n d ⟨ p, p ⟩ .$

Question Number 36910 Answers: 0 Comments: 0

1) decompose inside R(x) the fraction F(x)= (1/((1−x^2 )(1−x^3 ))) 2) find ∫ F(x)dx .

$1) d e c o m p o s e i n s i d e R (x) t h e f r a c t i o n$ $F (x) = \frac{1}{(1 - x^{2}) (1 - x^{3})}$ $2) f i n d \int F (x) d x .$

Question Number 36892 Answers: 1 Comments: 1

2. ∫[(√((1−x^2 )/(1+x^2 )))]dx=?

$2 . \int [\sqrt{(1 - x^{2}) / (1 + x^{2})}] d x = ?$

Question Number 36818 Answers: 1 Comments: 1

find f(a) = ∫ (dx/(√(1−ax^2 ))) with a from R .

$f i n d f (a) = \int \frac{d x}{\sqrt{1 - {a x}^{2}}} w i t h a f r o m R .$

Question Number 36811 Answers: 1 Comments: 0

∫ ((sin x)/(cos^2 x. (√(cos 2x)))) dx= ?

$\int \frac{\sin x}{\cos^{2} x . \sqrt{\cos 2 x}} d x = ?$

Question Number 36801 Answers: 2 Comments: 0

∫ ((1+x^4 )/((1−x^4 )^(3/2) )) dx = A ∫ A = B Find B ? Assume integration of constant=0.

$\int \frac{1 + x^{4}}{{(1 - x^{4})}^{\frac{3}{2}}} d x = A$ $\int A = B$ $Find B ?$ $Assume integration of constant = 0 .$

Question Number 36799 Answers: 1 Comments: 1

find ∫_0 ^∞ e^t ln(1+e^(−2t) )dt .

$f i n d \int_{0}^{\infty} e^{t} l n (1 + e^{- 2 t}) d t .$

Question Number 36762 Answers: 1 Comments: 2

find A_n = ∫_0 ^(π/4) (cosx +sinx)^n dx.

$f i n d A_{n} = \int_{0}^{\frac{π}{4}} {(c o s x + s i n x)}^{n} d x .$

Question Number 36755 Answers: 1 Comments: 4

let f(a) = ∫_0 ^1 e^t ln(1+ e^(−at) )dt with a≥0 1) find f(a) 2) calculate f^′ (a) 3) find the value of ∫_0 ^1 e^t ln(1+e^(−3t) )dt .

$l e t f (a) = \int_{0}^{1} e^{t} l n (1 + e^{- a t}) d t w i t h a ⩾ 0$ $1) f i n d f (a)$ $2) c a l c u l a t e f^{^{'}} (a)$ $3) f i n d t h e v a l u e o f \int_{0}^{1} e^{t} l n (1 + e^{- 3 t}) d t .$

Question Number 36754 Answers: 1 Comments: 1

calculate ∫_1 ^(+∞) (dx/(x^2 (√(4+x^2 )))) .

$c a l c u l a t e \int_{1}^{+ \infty} \frac{d x}{x^{2} \sqrt{4 + x^{2}}} .$

Question Number 36753 Answers: 1 Comments: 2

find I_n = ∫_0 ^1 x^n arctan(x)dx .

$f i n d I_{n} = \int_{0}^{1} x^{n} a r c t a n (x) d x .$

Question Number 36752 Answers: 1 Comments: 4

find ∫ (dx/(arcsinx(√(1−x^2 )))) .

$f i n d \int \frac{d x}{a r c s i n x \sqrt{1 - x^{2}}} .$

Question Number 36747 Answers: 0 Comments: 1

let f(x)= Σ_(n=1) ^∞ ((sin(nx))/n) x^n 1) prove that f is C^1 on ]−1,1[ 2)calculate f^′ (x) and prove that f(x)=arctan( ((xsinx)/(1−x cosx)))

$l e t f (x) = \sum_{n = 1}^{\infty} \frac{s i n (n x)}{n} x^{n}$ $1) p r o v e t h a t f i s C^{1} o n] - 1, 1 [$ $2) c a l c u l a t e f^{^{'}} (x) a n d p r o v e t h a t$ $f (x) = a r c t a n (\frac{x s i n x}{1 - x c o s x})$

Question Number 36738 Answers: 2 Comments: 3

(1) ∫(dα/((1+sin 2α)^2 ))= (2) ∫(dβ/((1+cos 2β)^2 ))= (3) ∫(dγ/((1+sin 2γ)(1+cos 2γ)))=

$(1) \int \frac{d α}{{(1 + \sin 2 α)}^{2}} =$ $(2) \int \frac{d β}{{(1 + \cos 2 β)}^{2}} =$ $(3) \int \frac{d γ}{(1 + \sin 2 γ) (1 + \cos 2 γ)} =$

Question Number 36737 Answers: 0 Comments: 1

let g(θ) =∫_0 ^1 ln( 1−e^(iθ) x^2 )dx find a simple form of g(θ) .θ from R.

$l e t g (θ) = \int_{0}^{1} l n (1 - e^{i θ} x^{2}) d x$ $f i n d a s i m p l e f o r m o f g (θ) . θ f r o m R .$

Question Number 36736 Answers: 0 Comments: 1

let f(θ) = ∫_0 ^1 ln(1−e^(iθ) x)dx find a simple form of f(θ)

$l e t f (θ) = \int_{0}^{1} l n (1 - e^{i θ} x) d x$ $f i n d a s i m p l e f o r m o f f (θ)$

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