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Question Number 103685 Answers: 1 Comments: 0

Question Number 103683 Answers: 1 Comments: 1

∫_0 ^1 tan^(−1) (((2x−1)/(1+x−x^2 )))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} {tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}{x}−\mathrm{1}}{\mathrm{1}+{x}−{x}^{\mathrm{2}} }\right){dx} \\ $$

Question Number 103673 Answers: 1 Comments: 0

Σ_(k=1) ^(4095) (1/(((√k)+(√(k+1)))((k)^(1/4) +((k+1))^(1/4) ))) ?

$$\underset{{k}=\mathrm{1}} {\overset{\mathrm{4095}} {\sum}}\frac{\mathrm{1}}{\left(\sqrt{{k}}+\sqrt{{k}+\mathrm{1}}\right)\left(\sqrt[{\mathrm{4}}]{{k}}+\sqrt[{\mathrm{4}}]{{k}+\mathrm{1}}\right)}\:? \\ $$

Question Number 103672 Answers: 1 Comments: 3

Question Number 103670 Answers: 4 Comments: 0

Given b_n = 3.2^n is a GP . find the value of (1/b_1 )+(1/b_2 )+(1/b_3 )+...+(1/b_(10) ) ?

$${Given}\:{b}_{{n}} \:=\:\mathrm{3}.\mathrm{2}^{{n}} \:{is}\:{a}\:{GP}\:.\:{find}\:{the}\:{value} \\ $$$${of}\:\frac{\mathrm{1}}{{b}_{\mathrm{1}} }+\frac{\mathrm{1}}{{b}_{\mathrm{2}} }+\frac{\mathrm{1}}{{b}_{\mathrm{3}} }+...+\frac{\mathrm{1}}{{b}_{\mathrm{10}} }\:?\: \\ $$

Question Number 103669 Answers: 2 Comments: 0

prove that : a) ∫_(−3) ^(−1) x^2 dx ≥∫_1 ^3 (2x−1)dx b)∫_(−2) ^0 xdx ≤∫_0 ^2 (x^2 + x )dx c)∫_1 ^4 (x^2 + 2)dx ≥∫_2 ^5 (2x −5)dx d)∫_(−π) ^(−((3π)/4)) cos 2x dx ≥∫_((3π)/4) ^π sin 2x dx

$${prove}\:{that}\:: \\ $$$$\left.{a}\right)\:\int_{−\mathrm{3}} ^{−\mathrm{1}} {x}^{\mathrm{2}} {dx}\:\geqslant\int_{\mathrm{1}} ^{\mathrm{3}} \left(\mathrm{2}{x}−\mathrm{1}\right){dx} \\ $$$$\left.{b}\right)\int_{−\mathrm{2}} ^{\mathrm{0}} {xdx}\:\leqslant\int_{\mathrm{0}} ^{\mathrm{2}} \left({x}^{\mathrm{2}} \:+\:{x}\:\right){dx} \\ $$$$\left.{c}\right)\int_{\mathrm{1}} ^{\mathrm{4}} \left({x}^{\mathrm{2}} \:+\:\mathrm{2}\right){dx}\:\:\geqslant\int_{\mathrm{2}} ^{\mathrm{5}} \left(\mathrm{2}{x}\:−\mathrm{5}\right){dx} \\ $$$$\left.{d}\right)\int_{−\pi} ^{−\frac{\mathrm{3}\pi}{\mathrm{4}}} \mathrm{cos}\:\mathrm{2}{x}\:{dx}\:\geqslant\int_{\frac{\mathrm{3}\pi}{\mathrm{4}}} ^{\pi} \mathrm{sin}\:\mathrm{2}{x}\:{dx} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 103665 Answers: 0 Comments: 0

Question Number 103664 Answers: 1 Comments: 0

Question Number 103659 Answers: 2 Comments: 1

When y=ax+b is a tangent line to the curve f(x)=x^3 passing through (0; −2), find a+b?

$$\mathrm{When}\:\mathrm{y}=\mathrm{ax}+\mathrm{b}\:\mathrm{is}\:\mathrm{a}\:\mathrm{tangent}\:\mathrm{line}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{curve}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{3}} \:\mathrm{passing}\:\mathrm{through}\:\left(\mathrm{0};\:−\mathrm{2}\right), \\ $$$$\mathrm{find}\:\mathrm{a}+\mathrm{b}? \\ $$

Question Number 103652 Answers: 0 Comments: 1

Question Number 103649 Answers: 0 Comments: 0

Question Number 103648 Answers: 2 Comments: 0

Evaluate (1/(1∙2∙3))+(3/(2∙3∙4))+(5/(3∙4∙5))+...+((2n−1)/(n(n+1)(n+2))

$$\boldsymbol{\mathrm{Evaluate}}\:\frac{\mathrm{1}}{\mathrm{1}\centerdot\mathrm{2}\centerdot\mathrm{3}}+\frac{\mathrm{3}}{\mathrm{2}\centerdot\mathrm{3}\centerdot\mathrm{4}}+\frac{\mathrm{5}}{\mathrm{3}\centerdot\mathrm{4}\centerdot\mathrm{5}}+...+\frac{\mathrm{2}\boldsymbol{\mathrm{n}}−\mathrm{1}}{\boldsymbol{\mathrm{n}}\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)\left(\boldsymbol{\mathrm{n}}+\mathrm{2}\right.} \\ $$

Question Number 103647 Answers: 0 Comments: 0

Question Number 103644 Answers: 1 Comments: 1

Question Number 103643 Answers: 4 Comments: 0

if sin x+cos x = (5/6) then (1/(sin x)) + (1/(cos x)) ?

$${if}\:\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\:=\:\frac{\mathrm{5}}{\mathrm{6}} \\ $$$${then}\:\frac{\mathrm{1}}{\mathrm{sin}\:{x}}\:+\:\frac{\mathrm{1}}{\mathrm{cos}\:{x}}\:?\: \\ $$

Question Number 103633 Answers: 2 Comments: 1

The question is Σ_(n=1) ^∞ (((2n−1)/(n(n+1)(n+2)))=...

$$ \\ $$$$\boldsymbol{\mathrm{The}}\:\boldsymbol{\mathrm{question}}\:\boldsymbol{\mathrm{is}} \\ $$$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{2}\boldsymbol{\mathrm{n}}−\mathrm{1}}{\boldsymbol{\mathrm{n}}\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)\left(\boldsymbol{\mathrm{n}}+\mathrm{2}\right.}\right)=... \\ $$

Question Number 103624 Answers: 0 Comments: 0

Question Number 103623 Answers: 2 Comments: 0

find the sum of the series whose nth term is ((2n−1)/(n(n+1)(n+2)). i have a problem with this and i need help please

Question Number 103622 Answers: 2 Comments: 0

Given a = Σ_(n=1) ^(24) (1/((√(n+1))+(√n))) then the value of a + (1/(log _a (bc)+1)) + (1/(log _b (ac)+1)) + (1/(log _c (ab)+1)) = ?

$${Given}\:{a}\:=\:\underset{{n}=\mathrm{1}} {\overset{\mathrm{24}} {\sum}}\frac{\mathrm{1}}{\sqrt{{n}+\mathrm{1}}+\sqrt{{n}}}\:{then}\:{the}\:{value}\:{of} \\ $$$${a}\:+\:\frac{\mathrm{1}}{\mathrm{log}\:_{{a}} \left({bc}\right)+\mathrm{1}}\:+\:\frac{\mathrm{1}}{\mathrm{log}\:_{{b}} \left({ac}\right)+\mathrm{1}}\:+ \\ $$$$\frac{\mathrm{1}}{\mathrm{log}\:_{{c}} \left({ab}\right)+\mathrm{1}}\:=\:? \\ $$

Question Number 103620 Answers: 1 Comments: 0

If a^2 −bc, b^2 −ac, c^2 −ab is AP where a+c = 12, find the value of a+b+c

$${If}\:{a}^{\mathrm{2}} −{bc},\:{b}^{\mathrm{2}} −{ac},\:{c}^{\mathrm{2}} −{ab}\:{is}\:{AP}\:{where}\:{a}+{c} \\ $$$$=\:\mathrm{12},\:{find}\:{the}\:{value}\:{of}\:{a}+{b}+{c}\: \\ $$

Question Number 103607 Answers: 1 Comments: 0

what is the value of ∫_c (x+2y)dx+(4−2x)dy around the ellipse C: (x^2 /(16))+(y^2 /8)=1 in the counterclockwise direction ?

$${what}\:{is}\:{the}\:{value}\:{of}\:\int_{{c}} \left({x}+\mathrm{2}{y}\right){dx}+\left(\mathrm{4}−\mathrm{2}{x}\right){dy} \\ $$$${around}\:{the}\:{ellipse}\:{C}:\:\frac{{x}^{\mathrm{2}} }{\mathrm{16}}+\frac{{y}^{\mathrm{2}} }{\mathrm{8}}=\mathrm{1} \\ $$$${in}\:{the}\:{counterclockwise} \\ $$$${direction}\:?\: \\ $$

Question Number 103606 Answers: 3 Comments: 1

an integer n between 1 and 98 , inclusive is to be chosen at random. what is the probability that n(n+1) will be divisible by 3

$${an}\:{integer}\:{n}\:{between}\:\mathrm{1}\:{and}\:\mathrm{98}\:, \\ $$$${inclusive}\:{is}\:{to}\:{be}\:{chosen}\:{at} \\ $$$${random}.\:{what}\:{is}\:{the}\:{probability} \\ $$$${that}\:{n}\left({n}+\mathrm{1}\right)\:{will}\:{be}\:{divisible}\:{by}\:\mathrm{3} \\ $$

Question Number 103603 Answers: 0 Comments: 1

Question Number 103597 Answers: 0 Comments: 2

pls help solve this differential equation (3x^2 sin ((1/x)) + y)dx = xcos((1/x)) −xdy

$$\boldsymbol{{pls}}\:\boldsymbol{{help}}\:\boldsymbol{{solve}}\:\boldsymbol{{this}}\:\boldsymbol{{differential}}\:\boldsymbol{{equation}} \\ $$$$\left(\mathrm{3}\boldsymbol{{x}}^{\mathrm{2}} \mathrm{sin}\:\left(\frac{\mathrm{1}}{\boldsymbol{{x}}}\right)\:+\:\boldsymbol{{y}}\right)\boldsymbol{{dx}}\:=\:\boldsymbol{{xcos}}\left(\frac{\mathrm{1}}{\boldsymbol{{x}}}\right)\:−\boldsymbol{{xdy}} \\ $$

Question Number 103593 Answers: 1 Comments: 0

calculate ∫_(−∞) ^∞ (dx/((x^2 +x +1)^2 (2x^2 +5)^2 ))

$$\mathrm{calculate}\:\:\int_{−\infty} ^{\infty} \:\:\:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{x}\:+\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{2x}^{\mathrm{2}} \:+\mathrm{5}\right)^{\mathrm{2}} } \\ $$

Question Number 103591 Answers: 1 Comments: 4

calculate ∫_3 ^(+∞) (dx/((x^2 −1)^3 (x+2)^2 ))

$$\mathrm{calculate}\:\:\int_{\mathrm{3}} ^{+\infty} \:\:\:\:\:\:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{3}} \left(\mathrm{x}+\mathrm{2}\right)^{\mathrm{2}} } \\ $$

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