Question and Answers Forum

All Questions Topic List

AllQuestion and Answers: Page 643

Question Number 150079 Answers: 2 Comments: 0

∫(dx/((x^2 +x+1)^2 ))

$\int \frac{d x}{{(x^{2} + x + 1)}^{2}}$

Question Number 150064 Answers: 1 Comments: 0

How can i evaluate the value of ∫_2 ^( 4) (e^t /t)dt = ?

$How can i evaluate the value of$ $\int_{2}^{4} \frac{e^{t}}{t} dt = ?$

Question Number 150063 Answers: 0 Comments: 0

Question Number 150062 Answers: 2 Comments: 0

Question Number 150059 Answers: 2 Comments: 0

Σ_(n=1) ^∞ (1/(n∙(2n + 1))) = ?

$\underset{n = 1}{\sum^{\infty}} \frac{1}{n \cdot (2 n + 1)} = ?$

Question Number 150058 Answers: 2 Comments: 2

Solve the equation: cos^4 (x) + i sin^4 (x) = 4e^(4ix)

$Solve the equation :$ $\cos^{4} (x) + i \sin^{4} (x) = {4 e}^{4 ix}$

Question Number 150056 Answers: 2 Comments: 1

Question Number 150053 Answers: 0 Comments: 0

let x;y;z;t>0 and x+y+z+t=4 prove that (4/((xyzt)^2 )) + 3 ≥ (√(45 + x^4 + y^4 + z^4 + t^4 ))

$let x; y; z; t > 0 and x + y + z + t = 4$ $prove that$ $\frac{4}{{(xyzt)}^{2}} + 3 ⩾ \sqrt{45 + x^{4} + y^{4} + z^{4} + t^{4}}$

Question Number 150045 Answers: 0 Comments: 0

Question Number 150044 Answers: 2 Comments: 0

Given f(((2x−3)/(2x+1)))+f(((2x+3)/(1−2x)))= 4x f(x)=?

$Given f (\frac{2 x - 3}{2 x + 1}) + f (\frac{2 x + 3}{1 - 2 x}) = 4 x$ $f (x) = ?$

Question Number 150040 Answers: 0 Comments: 0

Let a,b,c be positive real numbers such that a+b+c=1 .Prove that ((ab)/(1−c^2 )) +((bc)/(1−a^2 ))+((ca)/(1−b^2 )) ≤ (3/8)

$Let a, b, c be positive real numbers such$ $that a + b + c = 1 . Prove that$ $\frac{ab}{1 - c^{2}} + \frac{bc}{1 - a^{2}} + \frac{ca}{1 - b^{2}} ⩽ \frac{3}{8}$

Question Number 150039 Answers: 1 Comments: 0

Prove that (a/b)+(b/c)+(c/a)≥(√((a^2 +1)/(b^2 +1)))+(√((b^2 +1)/(c^2 +1)))+(√((c^2 +1)/(a^2 +1))) for a,b,c are positive real number

$Prove that \frac{a}{b} + \frac{b}{c} + \frac{c}{a} ⩾ \sqrt{\frac{a^{2} + 1}{b^{2} + 1}} + \sqrt{\frac{b^{2} + 1}{c^{2} + 1}} + \sqrt{\frac{c^{2} + 1}{a^{2} + 1}}$ $for a, b, c are positive real number$

Question Number 150037 Answers: 0 Comments: 0

Random Problem: ∫_(π/4) ^(π/2) (−7sin x + 3cos x) dx By getting the antiderivative of the trigonometric functions: ∫ sin(x) dx = −cos x + c ∫ cos(x) dx = sin x + c = −7 ∫ sin x + 3 ∫ cos x ∣_(π/4) ^(π/2) = −7(− cos x) + 3(sin x) ∣_(π/4) ^(π/2) = 7 cos x + 3sin x ∣_(π/4) ^(π/2) Evaluate it to the top and bottom limit of integration: = (7 cos ∙ (π/2) + 3 sin ∙ (π/2))− (7 cos ∙ (π/(4 )) + 3 sin ∙ (π/4) ) =[7(0) + 3(1)] − [7(((√2)/2)) + 3(((√2)/2))] = 3 − ((7(√2))/2) − ((3(√2))/2) = 3 − ((10(√2))/2) or 3 − 5(√2) Answer: 3 − 5(√2) Solution by Roswel:)

$R a n d o m P r o b l e m :$ $\underset{\frac{π}{4}}{\int^{\frac{π}{2}}} (- 7 \sin x + 3 \cos x) d x$ $B y g e t t i n g t h e a n t i d e r i v a t i v e o f t h e t r i g o n o m e t r i c f u n c t i o n s :$ $\int \sin (x) d x = - \cos x + c$ $\int \cos (x) d x = \sin x + c$ $= - 7 \int \sin x + 3 \int \cos x \underset{\frac{π}{4}}{\overset{\frac{π}{2}}{∣}} = - 7 (- \cos x) + 3 (\sin x) \underset{\frac{π}{4}}{\overset{\frac{π}{2}}{∣}}$ $= 7 \cos x + 3 \sin x \underset{\frac{π}{4}}{\overset{\frac{π}{2}}{∣}}$ $E v a l u a t e i t t o t h e t o p a n d b o t t o m l i m i t o f i n t e g r a t i o n :$ $= (7 \cos \cdot \frac{π}{2} + 3 \sin \cdot \frac{π}{2}) - (7 \cos \cdot \frac{π}{4} + 3 \sin \cdot \frac{π}{4})$ $= [7 (0) + 3 (1)] - [7 (\frac{\sqrt{2}}{2}) + 3 (\frac{\sqrt{2}}{2})]$ $= 3 - \frac{7 \sqrt{2}}{2} - \frac{3 \sqrt{2}}{2}$ $= 3 - \frac{10 \sqrt{2}}{2} o r 3 - 5 \sqrt{2}$ $A n s w e r : 3 - 5 \sqrt{2}$ $S o l u t i o n b y R o s w e l :)$

Question Number 150034 Answers: 1 Comments: 0

Find a closed form: a∈R and a≠0 Ω(a)=∫_( 0) ^( ∞) (x^4 /((1+x^2 )(1+a^4 x^4 ))) dx

$Find a closed form : a \in R and a \neq 0$ $Ω (a) = \underset{0}{\int^{\infty}} \frac{x^{4}}{(1 + x^{2}) (1 + a^{4} x^{4})} dx$

Question Number 150033 Answers: 1 Comments: 0

Question Number 150030 Answers: 0 Comments: 0

Roll a fair die twice and define A to be event that the sum of the scores showing up is greater than 7, B be the event that the sum of the scores showing up is a multiple of 3 and C be the event that the sum of the scores showing up is a prime number. Which of the events A,B and C are independent event? are the 3 events jointly independent?

$Roll a fair die twice and define A to be$ $event that the sum of the scores showing$ $up is greater than 7, B be the event that$ $the sum of the scores showing up is a$ $multiple of 3 and C be the event that$ $the sum of the scores showing up is a$ $prime number . Which of the events$ $A, B and C are independent event ?$ $are the 3 events jointly independent ?$