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IntegrationQuestion and Answers: Page 58

Question Number 158159 Answers: 2 Comments: 0

∫ (dx/(3−tan x)) =?

$\int \frac{d x}{3 - \tan x} = ?$

Question Number 158053 Answers: 2 Comments: 0

∫(dx/(sin^4 x))

$\int \frac{d x}{\sin^{4} x}$

Question Number 158009 Answers: 2 Comments: 0

Question Number 157977 Answers: 1 Comments: 0

∫ (dx/((1+(x)^(1/4) )(√x))) =?

$\int \frac{d x}{(1 + \sqrt[4]{x}) \sqrt{x}} = ?$

Question Number 157961 Answers: 2 Comments: 0

prove that: I=∫_0 ^( ∞) x^( 2) tanh(x).e^( −x) dx=(π^( 3) /8) −2

$p r o v e t h a t :$ $I = \int_{0}^{\infty} x^{2} t a n h (x) . e^{- x} d x = \frac{π^{3}}{8} - 2$

Question Number 157932 Answers: 0 Comments: 4

prove that tan^(−1) (((xy)/(rz)))+tan^(−1) (((xz)/(ry)))+tan^(−1) (((yz)/(rx)))=(π/2)

$p r o v e t h a t \tan^{- 1} (\frac{x y}{r z}) + \tan^{- 1} (\frac{x z}{r y}) + \tan^{- 1} (\frac{y z}{r x}) = \frac{π}{2}$

Question Number 157929 Answers: 0 Comments: 1

∫(dx/(sin x+ sec x)) using wiestress substitution

$\int \frac{dx}{\sin x + \sec x}$ $using wiestress substitution$

Question Number 157870 Answers: 1 Comments: 0

find the integral: ∫{(3x+1)/(x^2 +4)}dx

$f i n d t h e i n t e g r a l :$ $\int {(3 x + 1) / (x^{2} + 4)} d x$

Question Number 157826 Answers: 1 Comments: 0

Question Number 157823 Answers: 1 Comments: 0

calculate : Ω := Σ_(n=0) ^∞ (( 1)/((4n+1)^( 3) )) = ?

$c a l c u l a t e :$ $Ω := \underset{n = 0}{\sum^{\infty}} \frac{1}{{(4 n + 1)}^{3}} = ?$

Question Number 157744 Answers: 1 Comments: 0

∫_0 ^( 1) (( Li_( 2) (−x ) )/(1+ x))dx=?

$\int_{0}^{1} \frac{{Li}_{2} (- x)}{1 + x} dx = ?$

Question Number 157750 Answers: 1 Comments: 0

∫_0 ^∞ ((1−cos 4x)/(xe^x )) dx=?

$\int_{0}^{\infty} \frac{1 - \cos 4 x}{{x e}^{x}} d x = ?$

Question Number 157655 Answers: 1 Comments: 1

x^2 f(x^3 )+(1/((1+x)^2 )) f(((1−x)/(1+x)))=4x^3 (1+x^4 )^5 ∫_( 0) ^( 1) f(x) dx =?

$x^{2} f (x^{3}) + \frac{1}{{(1 + x)}^{2}} f (\frac{1 - x}{1 + x}) = 4 x^{3} {(1 + x^{4})}^{5}$ $\int_{0}^{1} f (x) d x = ?$

Question Number 157531 Answers: 2 Comments: 0

Question Number 157515 Answers: 0 Comments: 0

∫ ((x sin x)/(16x+9)) dx

$\int \frac{x \sin x}{16 x + 9} d x$

Question Number 157469 Answers: 1 Comments: 0

calculate : Ω:= ∫_(0 ) ^( 1) (( ln(1+x).ln(1−x))/(1+x)) dx =?

$c a l c u l a t e :$ $Ω := \int_{0}^{1} \frac{l n (1 + x) . l n (1 - x)}{1 + x} d x = ?$

Question Number 157456 Answers: 4 Comments: 0

∫ 5^(3−2x) dx =?

$\int 5^{3 - 2 x} d x = ?$

Question Number 157455 Answers: 1 Comments: 0

Question Number 157442 Answers: 2 Comments: 0

∫ (dx/(x−(√(x^2 +2x+2))))

$\int \frac{dx}{x - \sqrt{x^{2} + 2 x + 2}}$

Question Number 157412 Answers: 2 Comments: 0

∫_( 0) ^( (π/2)) (x/(sin^8 x+cos^8 x)) dx ?

$\int_{0}^{\frac{π}{2}} \frac{x}{\sin^{8} x + \cos^{8} x} d x ?$

Question Number 157309 Answers: 3 Comments: 0

∫ (dx/( (√x)+x(√(x+1)))) =?

$\int \frac{d x}{\sqrt{x} + x \sqrt{x + 1}} = ?$

Question Number 157268 Answers: 1 Comments: 0

prove that ∫(x^2 /((xsin x+cos x)^2 ))dx=−((xsec x)/(xsin x+cos x))+tan x+c

$p r o v e t h a t$ $\int \frac{x^{2}}{{(x \sin x + \cos x)}^{2}} d x = - \frac{x \sec x}{x \sin x + \cos x} + \tan x + c$

Question Number 157257 Answers: 1 Comments: 0

∫ ((sin^6 x+cos^5 x)/(sin^2 x cos^2 x)) dx

$\int \frac{\sin^{6} x + \cos^{5} x}{\sin^{2} x \cos^{2} x} d x$

Question Number 157251 Answers: 1 Comments: 0

# Nice Mathematics # ...calculation ... Ω :=∫_0 ^( 1) (( tanh^( −1) ((√( x)) ))/x) dx =^? (( π^( 2) )/4) −−−−−−−−−−−−− Ω :=^((√x) = t) 2∫_0 ^( 1) (( tanh^( −1) (t ))/t) dt :=^({tanh^( −1) (t )= (1/2) ln( ((1+t)/(1−t)) ) }) ∫_0 ^( 1) ((ln( 1+t )− ln(1−t ))/t) dt : = −Li_( 2) (−1 ) + Li_( 2) (1 ) :=^( {Li_( 2) (z )= Σ_(n=1) ^∞ (( z^( n) )/n^( 2) ) }) η (2) + ζ (2) := (π^( 2) /(12)) + (π^( 2) /6) = (( π^( 2) )/( 4)) ■ m.n

$You can't use 'macro parameter character #' in math mode$ $. . . c a l c u l a t i o n . . .$ $Ω := \int_{0}^{1} \frac{{t a n h}^{- 1} (\sqrt{x})}{x} d x \overset{?}{=} \frac{π^{2}}{4}$ $- - - - - - - - - - - - -$ $Ω : \overset{\sqrt{x} = t}{=} 2 \int_{0}^{1} \frac{{t a n h}^{- 1} (t)}{t} d t$ $: \overset{{{t a n h}^{- 1} (t) = \frac{1}{2} l n (\frac{1 + t}{1 - t})}}{=} \int_{0}^{1} \frac{l n (1 + t) - l n (1 - t)}{t} d t$ $: = - {Li}_{2} (- 1) + {Li}_{2} (1)$ $: \overset{{{Li}_{2} (z) = \underset{n = 1}{\sum^{\infty}} \frac{z^{n}}{n^{2}}}}{=} η (2) + ζ (2)$ $:= \frac{π^{2}}{12} + \frac{π^{2}}{6} = \frac{π^{2}}{4} ◼ m . n$

Question Number 157254 Answers: 1 Comments: 0

Show that ∫_0 ^1 (1/((x+1)(x+2)))dx = ln((4/3))

$Show that \int_{0}^{1} \frac{1}{(x + 1) (x + 2)} d x = \ln (\frac{4}{3})$

Question Number 157175 Answers: 2 Comments: 1

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