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IntegrationQuestion and Answers: Page 64

Question Number 152714 Answers: 0 Comments: 0

∫_0 ^( 1) ∫_0 ^( 1) ∫_0 ^( 1) ((x+y^2 +z^3 +1)/( (√(x+y+z))+1)) dxdydz

$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{x + y^{2} + z^{3} + 1}{\sqrt{x + y + z} + 1} d x d y d z$

Question Number 152703 Answers: 2 Comments: 0

∫_0 ^( 1) ∫_0 ^( 1) ∫_0 ^( 1) ((x+y^2 +z^3 )/(x+y+z)) dxdydz

$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{x + y^{2} + z^{3}}{x + y + z} d x d y d z$

Question Number 152701 Answers: 0 Comments: 0

∫_(−∞) ^( ∞) (1/( (√(x^2 +1)))) dx

$\int_{- \infty}^{\infty} \frac{1}{\sqrt{x^{2} + 1}} d x$

Question Number 152660 Answers: 1 Comments: 0

∫ (x/( (√(1 + x^3 )))) dx

$\int \frac{x}{\sqrt{1 + x^{3}}} dx$

Question Number 152580 Answers: 1 Comments: 0

Question Number 152576 Answers: 0 Comments: 1

∫ ((t + 13)/( ((t^2 + 5t + 6))^(1/3) )) dt

$\int \frac{t + 13}{\sqrt[3]{t^{2} + 5 t + 6}} dt$

Question Number 152572 Answers: 1 Comments: 0

∫_(−(Π/2)) ^(Π/2) ((1+cosx)/(3+2sinx))dx please,help me

$\int_{- \frac{Π}{2}}^{\frac{Π}{2}} \frac{1 + c o s x}{3 + 2 s i n x} d x$ $p l e a s e, h e l p m e$

Question Number 152533 Answers: 1 Comments: 0

Question Number 152502 Answers: 1 Comments: 0

∫ 9x^2 (4x^2 + 3)^(10) dx

$\int {9 x}^{2} {({4 x}^{2} + 3)}^{10} dx$

Question Number 152494 Answers: 1 Comments: 0

prove that :: Ω := ∫_(0 ) ^( ∞) (( e^( −x) .ln ((( 1)/( x)) ) sin ( x ))/(x )) dx = (( π)/( 8)) ( 2 γ +ln (2 ) ) ...■ m.n

$p r o v e t h a t ::$ $Ω := \int_{0}^{\infty} \frac{e^{- x} . \ln (\frac{1}{x}) s i n (x)}{x} d x = \frac{π}{8} (2 γ + \ln (2)) . . . ◼$ $m . n$

Question Number 152472 Answers: 1 Comments: 0

Γ((3/4)) = exp(− ((3γ)/4) + ∫_0 ^( 1) f(x)dx) find f(x)

$Γ (\frac{3}{4}) = \exp (- \frac{3 γ}{4} + \int_{0}^{1} f (x) d x)$ $find f (x)$

Question Number 152468 Answers: 1 Comments: 0

∫_(−∞) ^( ∞) ((ln((√(x^4 +1))))/( (√(x^4 +1)))) dx

$\int_{- \infty}^{\infty} \frac{\ln (\sqrt{x^{4} + 1})}{\sqrt{x^{4} + 1}} d x$

Question Number 152345 Answers: 1 Comments: 0

prove... S =Σ_(n=1) ^∞ (1/) = (1/3) +((2π (√3))/(27)) ...■

$p r o v e . . .$ $S = \underset{n = 1}{\sum^{\infty}} \frac{1}{} = \frac{1}{3} + \frac{2 π \sqrt{3}}{27} . . . ◼$

Question Number 152291 Answers: 1 Comments: 0

∫ (dx/(cos x+cosec x)) =?

$\int \frac{d x}{\cos x + cosec x} = ?$

Question Number 152276 Answers: 1 Comments: 0

Prove that ∫_0 ^π ((xtan x)/(sec x+tan x))dx=(π^2 /2)−π

$Prove that$ $\int_{0}^{π} \frac{xtan x}{\sec x + \tan x} dx = \frac{π^{2}}{2} - π$

Question Number 152275 Answers: 2 Comments: 0

∫_0 ^(π/2) sin 2xlog( tan x)dx

$\int_{0}^{\frac{π}{2}} \sin 2 xlog (\tan x) dx$

Question Number 152273 Answers: 1 Comments: 0

∫ ((tan θ+tan^3 θ)/(1+tan^3 θ))dθ

$\int \frac{\tan θ + \tan^{3} θ}{1 + \tan^{3} θ} d θ$

Question Number 152271 Answers: 1 Comments: 0

∫(3x−2)(√(x^2 +x+1)) dx

$\int (3 x - 2) \sqrt{x^{2} + x + 1} dx$

Question Number 152270 Answers: 2 Comments: 0

∫((5x+3)/( (√(x^2 +4x+10))))dx

$\int \frac{5 x + 3}{\sqrt{x^{2} + 4 x + 10}} dx$

Question Number 152492 Answers: 1 Comments: 0

A particle is projected upwards with a velocity of 96ms^(−1) . In addition to being subject to gravity, it is acted on by a retardation of 16t, where t is the time from the start of the motion. What is the greatest height attained by the particle?

$A particle is projected upwards with$ $a velocity of 96 {m s}^{- 1} . In addition to$ $being subject to gravity, it is acted on$ $by a retardation of 16 t, where t is the$ $time from the start of the motion .$ $What is the greatest height attained$ $by the particle ?$

Question Number 152203 Answers: 0 Comments: 1

If x is real show that (2+i)^((1+3i)x) +(2−i)^((1−3i)x) is also real

$If x is real show that$ ${(2 + i)}^{(1 + 3 i) x} + {(2 - i)}^{(1 - 3 i) x}$ $is also real$

Question Number 152201 Answers: 0 Comments: 0

...Integral... I := ∫_0 ^( π) ln (sin(x) ).tan^( −1) (cot(x))dx=^? 0 proof :: .... I := ∫_0 ^( π) ln (sin(x) ). tan^( −1) ( tan((π/2) −x ))dx := ∫_0 ^( π) ((π/2) −x ).ln(sin(x))dx := (π/2) ∫_0 ^( π) ln(sin(x))dx−∫_0 ^( π) xln(sin(x))dx := (π/2) (−π ln (2 )) −J ......( 1 ) J : = ∫_0 ^( π) (π − x) ln (sin(x))dx := π (−π ln(2))−J ∴ J :=((−π^( 2) )/2) ln( 2 ) .......(2) (2) ⇛ (1 ) : I = 0 .........■

$. . . Integral . . .$ $I := \int_{0}^{π} l n (s i n (x)) . {t a n}^{- 1} (c o t (x)) d x \overset{?}{=} 0$ $p r o o f :: . . . .$ $I := \int_{0}^{π} l n (s i n (x)) . {t a n}^{- 1} (t a n (\frac{π}{2} - x)) d x$ $:= \int_{0}^{π} (\frac{π}{2} - x) . l n (s i n (x)) d x$ $:= \frac{π}{2} \int_{0}^{π} l n (s i n (x)) d x - \int_{0}^{π} x l n (s i n (x)) d x$ $:= \frac{π}{2} (- π l n (2)) - J . . . . . . (1)$ $J : = \int_{0}^{π} (π - x) l n (s i n (x)) d x$ $:= π (- π l n (2)) - J$ $∴ J := \frac{- π^{2}}{2} l n (2) . . . . . . . (2)$ $(2) ⇛ (1) : I = 0 . . . . . . . . . ◼$

Question Number 152165 Answers: 2 Comments: 0

∫((sin x)/(sin 3x))dx

$\int \frac{\sin x}{\sin 3 x} dx$

Question Number 152164 Answers: 1 Comments: 2

∫((2x+1)/(4−3x−x^2 ))dx

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

Question Number 152163 Answers: 1 Comments: 0

∫(((x^2 +5x+3)/(x^2 +3x+2)))dx

$\int (\frac{x^{2} + 5 x + 3}{x^{2} + 3 x + 2}) dx$

Question Number 152161 Answers: 4 Comments: 0

∫cos (log x)dx

$\int \cos (\log x) dx$

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