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IntegrationQuestion and Answers: Page 57

Question Number 158517 Answers: 0 Comments: 0

ϑ =∫_( 0) ^( π/2) (dx/((1+(1/(sin^2 x)))^2 )) ?

$ϑ = \int_{0}^{π / 2} \frac{d x}{{(1 + \frac{1}{\sin^{2} x})}^{2}} ?$

Question Number 158590 Answers: 1 Comments: 0

prove that 1. I= ∫_0 ^( (π/2)) (( sin( x+tan(x)))/(sin(x)))dx =(π/2) 2. J = ∫_0 ^( (π/2)) ((sin(x−tan(x)))/(sin(x)))dx=((1/e) −(1/2))π

$p r o v e t h a t$ $1 . I = \int_{0}^{\frac{π}{2}} \frac{s i n (x + t a n (x))}{s i n (x)} d x = \frac{π}{2}$ $2 . J = \int_{0}^{\frac{π}{2}} \frac{s i n (x - t a n (x))}{s i n (x)} d x = (\frac{1}{e} - \frac{1}{2}) π$

Question Number 158494 Answers: 0 Comments: 0

Question Number 158492 Answers: 1 Comments: 0

∫ (dx/(1+x^7 )) ?

$\int \frac{dx}{1 + x^{7}} ?$

Question Number 158405 Answers: 2 Comments: 0

prove: ∫_0 ^∞ (1/(x^5 +x^4 +x^3 +x^2 +x+1))dx=(π/(3(√3)))

$p r o v e :$ $\int_{0}^{\infty} \frac{1}{x^{5} + x^{4} + x^{3} + x^{2} + x + 1} d x = \frac{π}{3 \sqrt{3}}$

Question Number 158402 Answers: 0 Comments: 0

∫_((1;π)) ^((2;π)) (1−(y^2 /x^2 )cos((y/x)))dx+(sin((y/x))+(y/x)cos((y/x)))dy=?

$\int_{(1; π)}^{(2; π)} (1 - \frac{y^{2}}{x^{2}} c o s (\frac{y}{x})) d x + (s i n (\frac{y}{x}) + \frac{y}{x} c o s (\frac{y}{x})) d y = ?$

Question Number 158420 Answers: 1 Comments: 0

∫_0 ^∞ ((lnx)/(1−x^2 ))dx

$\int_{0}^{\infty} \frac{lnx}{1 - x^{2}} dx$

Question Number 158333 Answers: 0 Comments: 0

∫_((1;π)) ^((2;π)) (1−(y^2 /x^2 )cos((y/x)))dx+(sin((y/x))+(y/x)cos((y/x)))dy=? OY on the road that doesn′t cut your arrow

$\int_{(1; π)}^{(2; π)} (1 - \frac{y^{2}}{x^{2}} \cos (\frac{y}{x})) dx + (\sin (\frac{y}{x}) + \frac{y}{x} \cos (\frac{y}{x})) dy = ?$ $OY on the road that {doesn}^{'} t cut your arrow$

Question Number 158320 Answers: 0 Comments: 0

prove that : Σ_(n=1) ^∞ (( H_( n) . F_n )/2^( n) ) = ln(4) + ((12)/( (√5))) ln( ϕ ) ϕ : Golden ratio F_( n) : fibonacci numbers

$p r o v e t h a t :$ $\underset{n = 1}{\sum^{\infty}} \frac{H_{n} . F_{n}}{2^{n}} = l n (4) + \frac{12}{\sqrt{5}} l n (φ)$ $φ : Golden ratio$ $F_{n} : f i b o n a c c i n u m b e r s$

Question Number 158288 Answers: 1 Comments: 0

Question Number 158230 Answers: 0 Comments: 0

∫(1/(1+ln x))dx=?

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

Question Number 158228 Answers: 1 Comments: 0

∫((5x^3 −3x^2 +7x−3)/((x^2 +1)^2 ))dx Solve by first finding the partial fraction

$\int \frac{5 x^{3} - 3 x^{2} + 7 x - 3}{{(x^{2} + 1)}^{2}} d x$ $S o l v e b y f i r s t f i n d i n g t h e p a r t i a l$ $f r a c t i o n$

Question Number 158209 Answers: 1 Comments: 0

Question Number 158207 Answers: 1 Comments: 0

Question Number 158190 Answers: 0 Comments: 1

∫((5x^3 −3x^2 +7x−3)/((x^2 +1)^2 ))dx Solve by first giving the partial functions

$\int \frac{5 x^{3} - 3 x^{2} + 7 x - 3}{{(x^{2} + 1)}^{2}} d x S o l v e b y$ $f i r s t g i v i n g t h e p a r t i a l f u n c t i o n s$

Question Number 158176 Answers: 1 Comments: 0

∫{((x^2 −x−21)/(2x^3 −x^2 +8x−4))}dx

$\int {\frac{x^{2} - x - 21}{2 x^{3} - x^{2} + 8 x - 4}} d x$

Question Number 158142 Answers: 0 Comments: 0

f(x)=x−[x] where [x] is the greatest integer function and −3≤x≤3 a) sketch f(x) b) state the domain of f(x) c) study the continuity of f(x) on its domain d) state the range of f(x)

$f (x) = x - [x] w h e r e [x] i s t h e g r e a t e s t$ $i n t e g e r f u n c t i o n a n d - 3 ⩽ x ⩽ 3$ $a) s k e t c h f (x)$ $b) s t a t e t h e d o m a i n o f f (x)$ $c) s t u d y t h e c o n t i n u i t y o f f (x) o n i t s d o m a i n$ $d) s t a t e t h e r a n g e o f f (x)$

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

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