Question and Answers Forum

let f(t) =∫_0 ^∞ ((cos^2 (tx))/((x^2 +3)^2 )) dx with t ≥0 1) give a explicit form of f(t) 2) find the value of ∫_0 ^∞ ((xsin(2tx))/((x^2 +3)^2 )) dx 3) give the values of integrals ∫_0 ^∞ (dx/((x^2 +3)^2 )) and ∫_0 ^∞ ((cos^2 (πx))/((x^2 +3)^2 ))dx 4) give the values of integrals ∫_0 ^∞ ((xsin(πx))/((x^2 +3)^2 )) and ∫_0 ^∞ ((xsin(((πx)/2)))/((x^2 +3)^2 )) dx .

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

Question Number 52683 Answers: 0 Comments: 3

let f(λ) =∫_(−∞) ^(+∞) ((sin(λx))/((x^2 +2λx +1)^2 ))dx with ∣λ∣<1 1) find the value of f(λ) 2) calculate ∫_(−∞) ^(+∞) ((sin((x/(2 ))))/((x^2 +x+1)^2 ))dx 3) find A(θ) =∫_(−∞) ^(+∞) ((sin((cosθ)x))/((x^2 +2cosθ x +1)^2 )) that we suppose 0<θ<(π/2)

$l e t f (λ) = \int_{- \infty}^{+ \infty} \frac{s i n (λ x)}{{(x^{2} + 2 λ x + 1)}^{2}} d x w i t h ∣ λ ∣< 1$ $1) f i n d t h e v a l u e o f f (λ)$ $2) c a l c u l a t e \int_{- \infty}^{+ \infty} \frac{s i n (\frac{x}{2})}{{(x^{2} + x + 1)}^{2}} d x$ $3) f i n d A (θ) = \int_{- \infty}^{+ \infty} \frac{s i n ((c o s θ) x)}{{(x^{2} + 2 c o s θ x + 1)}^{2}} t h a t w e s u p p o s e 0 < θ < \frac{π}{2}$

Question Number 52680 Answers: 0 Comments: 1

let f_n (x)=((sin(nx))/n^3 ) and f(x)=Σ_(n=1) ^∞ f_n (x) calculate ∫_0 ^π f(x)dx .

$l e t f_{n} (x) = \frac{s i n (n x)}{n^{3}} a n d f (x) = \sum_{n = 1}^{\infty} f_{n} (x)$ $c a l c u l a t e \int_{0}^{π} f (x) d x .$

Question Number 52667 Answers: 1 Comments: 0

∫((x^4 +1)/(x^2 (√(x^4 −1)))) dx

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

Question Number 52649 Answers: 0 Comments: 2

∫ ((4x^2 + 3)/((x^2 + x + 1)^2 )) dx

$\int \frac{{4 x}^{2} + 3}{{(x^{2} + x + 1)}^{2}} dx$

Question Number 52550 Answers: 1 Comments: 1

∫_0 ^( ∞) (x/(e^x − 1)) dx

$\int_{0}^{\infty} \frac{x}{e^{x} - 1} dx$

Question Number 52484 Answers: 2 Comments: 0

∫ ((cos x − x sin x)/(x cos x)) dx

$\int \frac{\cos x - x \sin x}{x \cos x} dx$

Question Number 52482 Answers: 0 Comments: 2

find the value or ∫_0 ^∞ ((arctan(x^2 ))/(1+x^4 ))dx .

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

Question Number 52459 Answers: 0 Comments: 2

let f(α)=∫_0 ^1 ((ln(1+iαx))/(1+x^2 ))dx 1)determine a explicit form of f(α) 2) calculate ∫_0 ^1 ((ln(1+ix))/(1+x^2 ))dx and ∫_0 ^1 ((ln(1+2ix))/(1+x^2 ))dx.

$l e t f (α) = \int_{0}^{1} \frac{l n (1 + i α x)}{1 + x^{2}} d x$ $1) d e t e r m i n e a e x p l i c i t f o r m o f f (α)$ $2) c a l c u l a t e \int_{0}^{1} \frac{l n (1 + i x)}{1 + x^{2}} d x a n d \int_{0}^{1} \frac{l n (1 + 2 i x)}{1 + x^{2}} d x .$