Question and Answers Forum

The values of a for which y= ax^2 +ax+(1/(24)) and x = ay^2 +ay+(1/(24)) touch each other are 1) (2/3) 2) (3/2) 3) ((13+(√(601)))/(12)) 4) ((13−(√(601)))/(12)).

$The values of a for which y = a x^{2} + a x + \frac{1}{24}$ $a n d x = {a y}^{2} + a y + \frac{1}{24} t o u c h e a c h o t h e r$ $a r e$ $1) \frac{2}{3} 2) \frac{3}{2}$ $3) \frac{13 + \sqrt{601}}{12} 4) \frac{13 - \sqrt{601}}{12} .$

Question Number 39383 Answers: 1 Comments: 1

calculate ∫_0 ^(π/3) ((sinxdx)/(cosx(2+ln(cosx))) .

$c a l c u l a t e \int_{0}^{\frac{π}{3}} \frac{s i n x d x}{c o s x (2 + l n (c o s x)} .$

Question Number 39382 Answers: 1 Comments: 0

Question Number 39381 Answers: 1 Comments: 0

Question Number 39380 Answers: 0 Comments: 0

find the value of Σ_(n=0) ^∞ (((−1)^n )/(4n+1))

$f i n d t h e v a l u e o f \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{4 n + 1}$

Question Number 39379 Answers: 1 Comments: 6

Question Number 39378 Answers: 0 Comments: 1

study the derivability of f(x)=Σ_(n=0) ^∞ (((−1)^n )/(nx +1))

$s t u d y t h e d e r i v a b i l i t y o f$ $f (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n x + 1}$

Question Number 39376 Answers: 0 Comments: 0

how to calculate the product (Σ_(n=0) ^∞ a_n x^n ).(Σ_(n=0) ^∞ b_n x^(2n) )?

$h o w t o c a l c u l a t e t h e p r o d u c t (\sum_{n = 0}^{\infty} a_{n} x^{n}) . (\sum_{n = 0}^{\infty} b_{n} x^{2 n}) ?$

Question Number 39375 Answers: 0 Comments: 1

Question Number 39374 Answers: 0 Comments: 2

calculate I = ∫_0 ^∞ ((arctan(x^2 ))/(1+x^2 ))dx

$c a l c u l a t e I = \int_{0}^{\infty} \frac{a r c t a n (x^{2})}{1 + x^{2}} d x$

Question Number 39373 Answers: 0 Comments: 1

find the values of integrals A = ∫_(−∞) ^(+∞) cos(x^2 +x+1)dx and B = ∫_(−∞) ^(+∞) sin(x^2 +x+1)dx

$f i n d t h e v a l u e s o f i n t e g r a l s$ $A = \int_{- \infty}^{+ \infty} c o s (x^{2} + x + 1) d x a n d B = \int_{- \infty}^{+ \infty} s i n (x^{2} + x + 1) d x$

Question Number 39371 Answers: 2 Comments: 1

Question Number 39370 Answers: 0 Comments: 1

let I (λ) = ∫_(−∞) ^(+∞) ((cos(λx))/((1+ix)^2 ))dx 1) extract Re(I(λ)) and Im(I(λ)) 2) calculate I(λ) 3) conclude the values of Re(I(λ)) and Im(I(λ)).

$l e t I (λ) = \int_{- \infty}^{+ \infty} \frac{c o s (λ x)}{{(1 + i x)}^{2}} d x$ $1) e x t r a c t R e (I (λ)) a n d I m (I (λ))$ $2) c a l c u l a t e I (λ)$ $3) c o n c l u d e t h e v a l u e s o f R e (I (λ)) a n d I m (I (λ)) .$

Question Number 39369 Answers: 0 Comments: 1

1) calculate F(x)= ∫_1 ^(√x) ((arctan(t))/t^2 )dt with x≥1 2) calculate A_n = ∫_1 ^(√n) ((arctan(t))/t^2 ) dt and find lim_(n→+∞) A_n

$1) c a l c u l a t e F (x) = \int_{1}^{\sqrt{x}} \frac{a r c t a n (t)}{t^{2}} d t w i t h x ⩾ 1$ $2) c a l c u l a t e A_{n} = \int_{1}^{\sqrt{n}} \frac{a r c t a n (t)}{t^{2}} d t a n d f i n d {l i m}_{n \to + \infty} A_{n}$

Question Number 39368 Answers: 0 Comments: 1

let F(t)= ∫_0 ^(+∞) ((sinx)/(x(1+x^2 ))) e^(−tx(1+x^2 )) dx witht≥0 1) caculate (dF/dt)(t) 2) find a simple form of F(t) 3) find the value of ∫_0 ^∞ ((sinx)/(x(1+x^2 )dx )).

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

Question Number 39365 Answers: 0 Comments: 1

Question Number 39357 Answers: 0 Comments: 0

∫ (1/(xln(x+1))) dx

$\int \frac{1}{x l n (x + 1)} d x$

Question Number 39349 Answers: 1 Comments: 1

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