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IntegrationQuestion and Answers: Page 239

Question Number 57948 Answers: 0 Comments: 0

let A(ξ) =∫_ξ ^ξ^2 ((arctan(1+ξt)−(π/4))/((√(2+ξt))−(√(2−ξt)))) dt find lim_(ξ →0) A(ξ) .

$l e t A (ξ) = \int_{ξ}^{ξ^{2}} \frac{a r c t a n (1 + ξ t) - \frac{π}{4}}{\sqrt{2 + ξ t} - \sqrt{2 - ξ t}} d t$ $f i n d {l i m}_{ξ \to 0} A (ξ) .$

Question Number 57900 Answers: 0 Comments: 1

let f(x) =∫_0 ^∞ ((cos(πxt))/((t^2 +3x^2 )^2 )) dt with x>0 1) find a explicit form for f(x) 2) find the value of ∫_0 ^∞ ((cos(πt))/((t^2 +3)^2 ))dt 3) let U_n =f(n) find nature of Σ U_n

$l e t f (x) = \int_{0}^{\infty} \frac{c o s (π x t)}{{(t^{2} + 3 x^{2})}^{2}} d t w i t h x > 0$ $1) f i n d a e x p l i c i t f o r m f o r f (x)$ $2) f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{c o s (π t)}{{(t^{2} + 3)}^{2}} d t$ $3) l e t U_{n} = f (n) f i n d n a t u r e o f Σ U_{n}$

Question Number 57899 Answers: 0 Comments: 2

let f(x) =∫_0 ^(+∞) (dt/((t^2 +x^2 )^3 )) with x>0 1) find a explicit form off (x) 1) calculate ∫_0 ^∞ (dx/((t^2 +3)^3 )) and ∫_0 ^∞ (dt/((t^2 +4)^3 )) 2) find the value of A(θ) =∫_0 ^∞ (dt/((t^2 +sin^2 θ)^3 )) with 0<θ<π.

$l e t f (x) = \int_{0}^{+ \infty} \frac{d t}{{(t^{2} + x^{2})}^{3}} w i t h x > 0$ $1) f i n d a e x p l i c i t f o r m o f f (x)$ $1) c a l c u l a t e \int_{0}^{\infty} \frac{d x}{{(t^{2} + 3)}^{3}} a n d \int_{0}^{\infty} \frac{d t}{{(t^{2} + 4)}^{3}}$ $2) f i n d t h e v a l u e o f A (θ) = \int_{0}^{\infty} \frac{d t}{{(t^{2} + {s i n}^{2} θ)}^{3}} w i t h 0 < θ < π .$

Question Number 57825 Answers: 1 Comments: 1

calculate ∫_0 ^π ((2xsinx)/(3 +cos(2x)))dx .

$c a l c u l a t e \int_{0}^{π} \frac{2 x s i n x}{3 + c o s (2 x)} d x .$

Question Number 57821 Answers: 2 Comments: 0

find the common area of: { (((x^2 /3)+y^2 =1)),((x^2 +(y^2 /3)=1)) :}

$find the common area of :$ ${\begin{cases} \frac{x^{2}}{3} + y^{2} = 1 \\ x^{2} + \frac{y^{2}}{3} = 1 \end{cases}$

Question Number 57819 Answers: 2 Comments: 7

a. ∫ [((1−e^x )/(1+e^x ))]^(1/2) dx=? b. ∫ ((lnx)/(√(1+x)))=? c. ∫_( (√e)) ^( e) sin(lnx)dx=?

$a . \int [\frac{1 - e^{x}}{1 + e^{x}}]^{\frac{1}{2}} dx = ?$ $b . \int \frac{lnx}{\sqrt{1 + x}} = ?$ $c . \underset{\sqrt{e}}{\int^{e}} \sin (lnx) dx = ?$

Question Number 57817 Answers: 1 Comments: 1

find the value of ∫_(π/3) ^(π/2) (dx/(√(2cos^2 x +3sin^2 x)))

$f i n d t h e v a l u e o f \int_{\frac{π}{3}}^{\frac{π}{2}} \frac{d x}{\sqrt{2 {c o s}^{2} x + 3 {s i n}^{2} x}}$

Question Number 57750 Answers: 1 Comments: 0

find ∫ x^2 (√(25−x^2 ))dx

$f i n d \int x^{2} \sqrt{25 - x^{2}} d x$

Question Number 57749 Answers: 1 Comments: 3

find ∫ (dx/(x^2 (√(9+x^2 ))))

$f i n d \int \frac{d x}{x^{2} \sqrt{9 + x^{2}}}$

Question Number 57748 Answers: 2 Comments: 2

find ∫ x^2 (√(4+x^2 ))dx

$f i n d \int x^{2} \sqrt{4 + x^{2}} d x$

Question Number 57746 Answers: 0 Comments: 4

let f(x)=∫_(−∞) ^(+∞) (dt/((t^2 −2xt +1)^2 )) with ∣x∣<1 (x real) 1) determine a explicit form for f(x) 2) find also g(x) =∫_(−∞) ^(+∞) ((tdt)/((t^2 −2xt +1)^3 )) 3) calculate ∫_(−∞) ^(+∞) (dt/((t^2 −(√2)t +1)^2 )) and ∫_(−∞) ^(+∞) ((tdt)/((t^2 −(√2)t +1)^3 )) 4) calculate A(θ) =∫_(−∞) ^(+∞) (dt/((t^2 −2cosθ t+1)^2 )) and B(θ) =∫_(−∞) ^(+∞) ((tdt)/((t^2 −2cosθ t +1)^3 )) with 0<θ <(π/2) .

$l e t f (x) = \int_{- \infty}^{+ \infty} \frac{d t}{{(t^{2} - 2 x t + 1)}^{2}} w i t h ∣ x ∣< 1 (x r e a l)$ $1) d e t e r m i n e a e x p l i c i t f o r m f o r f (x)$ $2) f i n d a l s o g (x) = \int_{- \infty}^{+ \infty} \frac{t d t}{{(t^{2} - 2 x t + 1)}^{3}}$ $3) c a l c u l a t e \int_{- \infty}^{+ \infty} \frac{d t}{{(t^{2} - \sqrt{2} t + 1)}^{2}} a n d \int_{- \infty}^{+ \infty} \frac{t d t}{{(t^{2} - \sqrt{2} t + 1)}^{3}}$ $4) c a l c u l a t e A (θ) = \int_{- \infty}^{+ \infty} \frac{d t}{{(t^{2} - 2 c o s θ t + 1)}^{2}} a n d$ $B (θ) = \int_{- \infty}^{+ \infty} \frac{t d t}{{(t^{2} - 2 c o s θ t + 1)}^{3}} w i t h 0 < θ < \frac{π}{2} .$

Question Number 57668 Answers: 0 Comments: 3

let V_n = ∫_0 ^(1+(1/n)) ((x+1)/(√(2x^2 +3))) dx 1) calculate lim_(n→+∞) V_n 2) find nature of the serie Σ V_n

$l e t V_{n} = \int_{0}^{1 + \frac{1}{n}} \frac{x + 1}{\sqrt{2 x^{2} + 3}} d x$ $1) c a l c u l a t e {l i m}_{n \to + \infty} V_{n}$ $2) f i n d n a t u r e o f t h e s e r i e Σ V_{n}$

Question Number 57667 Answers: 0 Comments: 3

calculate U_n =∫_(π/n) ^((2π)/n) (dx/(2 +sinx)) 1) calculate U_n and lim_(n→+∞) nU_n 2) find nature of Σ U_n

$c a l c u l a t e U_{n} = \int_{\frac{π}{n}}^{\frac{2 π}{n}} \frac{d x}{2 + s i n x}$ $1) c a l c u l a t e U_{n} a n d {l i m}_{n \to + \infty} {n U}_{n}$ $2) f i n d n a t u r e o f Σ U_{n}$

Question Number 57666 Answers: 0 Comments: 3

1) calculate f(θ) =∫_0 ^1 (√(t^2 +2sinθt +1))dt with 0≤θ≤(π/2) 2) calculate g(t) =∫_0 ^1 (√(t^2 +2(sinθ)t +1))dθ 3) find also h(θ) =∫_0 ^1 (t/(√(t^2 +2(sinθ)t +1)))dt

$1) c a l c u l a t e f (θ) = \int_{0}^{1} \sqrt{t^{2} + 2 s i n θ t + 1} d t w i t h 0 ⩽ θ ⩽ \frac{π}{2}$ $2) c a l c u l a t e g (t) = \int_{0}^{1} \sqrt{t^{2} + 2 (s i n θ) t + 1} d θ$ $3) f i n d a l s o h (θ) = \int_{0}^{1} \frac{t}{\sqrt{t^{2} + 2 (s i n θ) t + 1}} d t$

Question Number 57665 Answers: 0 Comments: 4

let f(a) =∫_(π/4) ^(π/3) (√(a+tan^2 x))dx with a>0 1) find a explicit form of f(a) 2) find also g(a) =∫_(π/4) ^(π/3) (dx/(√(a+tan^2 x))) 3) find the values of ∫_(π/4) ^(π/3) (√(2+tan^2 x))dx and ∫_(π/4) ^(π/3) (dx/(√(3+tan^2 x)))

$l e t f (a) = \int_{\frac{π}{4}}^{\frac{π}{3}} \sqrt{a + {t a n}^{2} x} d x w i t h a > 0$ $1) f i n d a e x p l i c i t f o r m o f f (a)$ $2) f i n d a l s o g (a) = \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{d x}{\sqrt{a + {t a n}^{2} x}}$ $3) f i n d t h e v a l u e s o f \int_{\frac{π}{4}}^{\frac{π}{3}} \sqrt{2 + {t a n}^{2} x} d x a n d \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{d x}{\sqrt{3 + {t a n}^{2} x}}$