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IntegrationQuestion and Answers: Page 76

Question Number 142656 Answers: 2 Comments: 0

∫_(−π) ^π ((xsin x)/(1+x^2 ))dx=?

$\int_{- π}^{π} \frac{xsin x}{1 + x^{2}} dx = ?$

Question Number 142643 Answers: 1 Comments: 0

∫_0 ^(π/2) ((cos^2 t)/(sint))dt

$\int_{0}^{\frac{π}{2}} \frac{{c o s}^{2} t}{s i n t} d t$

Question Number 142629 Answers: 1 Comments: 0

Σ_(n=1) ^∞ (−1)^n ∙((2n−1)/((2n)!))∙((π/2))^(2n) =Σ_(n=1) ^∞ ((2n−1)/((2n)!))∙(−((π/2))^2 )^n =(2xD−1)∣_(x=π/2) Σ_(n=1) ^∞ (((−x^2 )^n )/((2n)!)) =(2xD−1)∣_(x=π/2) [Σ_(n=0) ^∞ (((−x^2 )^n )/((2n)!))−1] =(2xD−1)∣_(x=π/2) (cos x−1) =(−2xsin x−cos x+1)∣_(x=π/2) =1−π where is wrong?

$\underset{n = 1}{\sum^{\infty}} {(- 1)}^{n} \cdot \frac{2 n - 1}{(2 n)!} \cdot {(\frac{π}{2})}^{2 n}$ $= \underset{n = 1}{\sum^{\infty}} \frac{2 n - 1}{(2 n)!} \cdot {(- {(\frac{π}{2})}^{2})}^{n}$ $= (2 xD - 1) ∣_{x = π / 2} \underset{n = 1}{\sum^{\infty}} \frac{{(- x^{2})}^{n}}{(2 n)!}$ $= (2 xD - 1) ∣_{x = π / 2} [\underset{n = 0}{\sum^{\infty}} \frac{{(- x^{2})}^{n}}{(2 n)!} - 1]$ $= (2 xD - 1) ∣_{x = π / 2} (\cos x - 1)$ $= (- 2 xsin x - \cos x + 1) ∣_{x = π / 2}$ $= 1 - π$ $where is wrong ?$

Question Number 142595 Answers: 1 Comments: 0

Prove :: sec^2 x=4Σ_(n=0) ^∞ {(1/([(2n+1)π−2x]^2 ))+(1/([(2n+1)π+2x]^2 ))}

$Prove :: \sec^{2} x = 4 \underset{n = 0}{\sum^{\infty}} {\frac{1}{{[(2 n + 1) π - 2 x]}^{2}} + \frac{1}{{[(2 n + 1) π + 2 x]}^{2}}}$

Question Number 142573 Answers: 2 Comments: 0

Question Number 142545 Answers: 1 Comments: 0

nice .....integral Ω:=∫_(−∞) ^( +∞) (dx/((x^2 +π^2 )cosh(x))) =? .....

$n i c e . . . . . i n t e g r a l$ $Ω := \int_{- \infty}^{+ \infty} \frac{d x}{(x^{2} + π^{2}) c o s h (x)} = ?$ $. . . . .$

Question Number 142516 Answers: 0 Comments: 0

∫(dx/((−lnx)^(1/x) ))

$\int \frac{d x}{{(- l n x)}^{\frac{1}{x}}}$

Question Number 142505 Answers: 0 Comments: 0

Question Number 142504 Answers: 1 Comments: 0

Question Number 142488 Answers: 2 Comments: 0

∫_0 ^1 (t^(k−1) /(1+t^2 ))dt

$\int_{0}^{1} \frac{t^{k - 1}}{1 + t^{2}} d t$

Question Number 142475 Answers: 2 Comments: 0

Question Number 142469 Answers: 1 Comments: 0

...... Calculus ..... Evaluate: ∫_0 ^( 1) (((log((1/x)))/(1−x)))^3 dx=??

$. . . . . . C a l c u l u s . . . . .$ $E v a l u a t e : \int_{0}^{1} {(\frac{l o g (\frac{1}{x})}{1 - x})}^{3} d x = ? ?$

Question Number 142457 Answers: 1 Comments: 0

I=∫(dx/( (√(a^2 −(x+(1/x))))))

$I = \int \frac{d x}{\sqrt{a^{2} - (x + \frac{1}{x})}}$

Question Number 142461 Answers: 2 Comments: 1

Question Number 142438 Answers: 1 Comments: 0

∫_0 ^1 ((1−x)/(lnx))dx how many tricks solve this

$\int_{0}^{1} \frac{1 - x}{l n x} d x$ $h o w m a n y t r i c k s s o l v e t h i s$

Question Number 142362 Answers: 1 Comments: 0

prove that: ∫_0 ^( ∞) ln((1/x)).j_0 (x)dx:= γ+ln(2) Hint:(1) j_0 (x)=Σ_(n=0) ^∞ (((−1)^n x^(2n) )/(2^(2n) .Γ^2 (n+1))) (Bessel function) Hint:2 L [ j_0 (x)]=(1/( (√(1+s^2 ))))

$p r o v e t h a t :$ $\int_{0}^{\infty} l n (\frac{1}{x}) . j_{0} (x) d x := γ + l n (2)$ $H i n t : (1)$ $j_{0} (x) = \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n} x^{2 n}}{2^{2 n} . Γ^{2} (n + 1)} (B e s s e l f u n c t i o n)$ $H i n t : 2$ $L [j_{0} (x)] = \frac{1}{\sqrt{1 + s^{2}}}$