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IntegrationQuestion and Answers: Page 209

Question Number 69784 Answers: 0 Comments: 0

calculate f(a) =∫_0 ^∞ e^(−(x^2 +(a/x^2 ))) dx with a>0

$c a l c u l a t e f (a) = \int_{0}^{\infty} e^{- (x^{2} + \frac{a}{x^{2}})} d x w i t h a > 0$

Question Number 69692 Answers: 0 Comments: 0

Question Number 69637 Answers: 1 Comments: 0

...now try this one: ∫(dx/(x^(1/2) −x^(1/3) −x^(1/6) ))=

$. . . now try this one :$ $\int \frac{d x}{x^{1 / 2} - x^{1 / 3} - x^{1 / 6}} =$

Question Number 69623 Answers: 1 Comments: 0

∫(1/((√x) + (x)^(1/3) )) dx

$\int \frac{1}{\sqrt{x} + \sqrt[3]{x}} d x$

Question Number 69603 Answers: 1 Comments: 0

∫ x^3 arcsinxdx

$\int x^{3} a r c s i n x d x$

Question Number 69597 Answers: 1 Comments: 1

Question Number 69573 Answers: 0 Comments: 1

Question Number 69570 Answers: 0 Comments: 0

Question Number 69566 Answers: 1 Comments: 0

Question Number 69565 Answers: 0 Comments: 0

Question Number 69564 Answers: 0 Comments: 3

let f(a) =∫_0 ^∞ (dx/(x^4 −2x^2 +a)) with a real and a>1 1) determine a explicit form for f(a) 2) calculate g(a) =∫_0 ^∞ (dx/((x^4 −2x^2 +a)^2 )) 3) find the values of integrals ∫_0 ^∞ (dx/(x^4 −2x^2 +3)) and ∫_0 ^∞ (dx/((x^4 −2x^2 +3)^2 ))

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

Question Number 69563 Answers: 0 Comments: 1

calculate ∫_0 ^∞ (dx/(x^4 −x^2 +1))

$c a l c u l a t e \int_{0}^{\infty} \frac{d x}{x^{4} - x^{2} + 1}$

Question Number 69502 Answers: 3 Comments: 2

∫((3sinx+4cosx)/(4sinx+3cosx))dx

$\int \frac{3 sinx + 4 cosx}{4 sinx + 3 cosx} dx$

Question Number 69390 Answers: 0 Comments: 0

study the convergence of ∫_0 ^∞ (((1+x)^α −(1+x)^((β) )/x)dx and determine its value

$s t u d y t h e c o n v e r g e n c e o f$ $\int_{0}^{\infty} \frac{{(1 + x)}^{α} - {(1 + x)}^{(β}}{x} d x$ $a n d d e t e r m i n e i t s v a l u e$

Question Number 69389 Answers: 0 Comments: 0

find ∫_(∣z+i∣=3) ((sinz)/(z+i))dz

$f i n d \int_{∣ z + i ∣= 3} \frac{s i n z}{z + i} d z$

Question Number 69379 Answers: 0 Comments: 1

calculste f(a) =∫_(−∞) ^(+∞) (((−1)^x^2 )/(x^2 +a^2 ))dx with a>0

$c a l c u l s t e f (a) = \int_{- \infty}^{+ \infty} \frac{{(- 1)}^{x^{2}}}{x^{2} + a^{2}} d x w i t h a > 0$

Question Number 69375 Answers: 0 Comments: 0

let f(α) =∫_0 ^∞ ((cos(α(1+x^2 )))/(1+x^2 ))dx 1)determine a explicit form of f(α) 2) calculate ∫_0 ^∞ ((cos(2+2x^2 ))/(x^2 +1))dx

$l e t f (α) = \int_{0}^{\infty} \frac{c o s (α (1 + x^{2}))}{1 + x^{2}} d x$ $1) d e t e r m i n e a e x p l i c i t f o r m o f f (α)$ $2) c a l c u l a t e \int_{0}^{\infty} \frac{c o s (2 + 2 x^{2})}{x^{2} + 1} d x$

Question Number 74637 Answers: 1 Comments: 1

1)calculate f(x)=∫_0 ^1 t^2 (√(x^2 +t^2 ))dt with x>0 2) calculste g(x)=∫_0 ^1 (t^2 /(√(x^2 +t^2 )))dt

$1) c a l c u l a t e f (x) = \int_{0}^{1} t^{2} \sqrt{x^{2} + t^{2}} d t w i t h x > 0$ $2) c a l c u l s t e g (x) = \int_{0}^{1} \frac{t^{2}}{\sqrt{x^{2} + t^{2}}} d t$

Question Number 69261 Answers: 1 Comments: 0

find ∫_0 ^1 xtanx dx

$f i n d \int_{0}^{1} x t a n x d x$

Question Number 69257 Answers: 0 Comments: 0

∫_(−2) ^( 4) ∣x∣^(2x^3 ) dx = ?

$\int_{- 2}^{4} ∣ x ∣^{2 x^{3}} d x = ?$

Question Number 69246 Answers: 0 Comments: 0

Explicit ∫_0 ^∞ ((Si(ax))/(x+b)) dx with Si(u)=∫_0 ^u ((sinx)/x)dx

$E x p l i c i t \int_{0}^{\infty} \frac{S i (a x)}{x + b} d x w i t h S i (u) = \int_{0}^{u} \frac{s i n x}{x} d x$

Question Number 69241 Answers: 0 Comments: 0

Let consider K=∫_0 ^1 (((1−x^a )(1−x^b )(1−x^c ))/((x−1)lnx))dx prove that e^K = (((a+b)!(a+c)!(b+c)!)/(a!b!c!(a+b+c)!))

$L e t c o n s i d e r K = \int_{0}^{1} \frac{(1 - x^{a}) (1 - x^{b}) (1 - x^{c})}{(x - 1) l n x} d x$ $p r o v e t h a t$ $e^{K} = \frac{(a + b)! (a + c)! (b + c)!}{a! b! c! (a + b + c)!}$

Question Number 69238 Answers: 1 Comments: 1

Use Residus Theorem to explicit f(a)=Σ_(n=1) ^∞ (((−1)^n sin(na))/n^3 )

$U s e R e s i d u s T h e o r e m t o e x p l i c i t$ $f (a) = \underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n} s i n (n a)}{n^{3}}$

Question Number 69233 Answers: 1 Comments: 1

Prove that B=∫_0 ^1 [ln(−lnu)]^2 du = γ^2 + ζ(2)

$P r o v e t h a t B = \int_{0}^{1} {[l n (- l n u)]}^{2} d u = γ^{2} + ζ (2)$

Question Number 69231 Answers: 0 Comments: 1

Prove that Σ_(p=0) ^∞ (((−1)^p )/(4p+1)) = ((π−argcoth((√2) ))/(4(√2))) and Σ_(p=0) ^(∞ ) (((−1)^p )/(4p+3)) = ((π+argcoth((√2) ))/(4(√2) ))

$P r o v e t h a t \underset{p = 0}{\sum^{\infty}} \frac{{(- 1)}^{p}}{4 p + 1} = \frac{π - a r g c o t h (\sqrt{2})}{4 \sqrt{2}} a n d$ $\underset{p = 0}{\sum^{\infty}} \frac{{(- 1)}^{p}}{4 p + 3} = \frac{π + a r g c o t h (\sqrt{2})}{4 \sqrt{2}}$

Question Number 69167 Answers: 1 Comments: 0

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