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IntegrationQuestion and Answers: Page 244

Question Number 54102 Answers: 1 Comments: 3

the absolute value ∫_(10) ^(19) ((cos x)/(1+x^8 )) dx is...

$the absolute value$ $\int_{10}^{19} \frac{\cos x}{1 + x^{8}} d x is . . .$

Question Number 54074 Answers: 1 Comments: 5

Evaluate : 1) ∫_0 ^( 1) (dx/((√(1+x))+(√(1−x))+2)) 2) ∫_0 ^( 2) ((ln(1+2x))/(1+x^2 )) 3) ∫_0 ^( π) (x/(√(1+sin^3 x)))((3πcosx+4sinx)sin^2 x+4)dx 4) ∫_0 ^( π) ((x^2 cos^2 x−xsinx−cosx−1)/((1+xsinx)^2 )) dx.

$E v a l u a t e :$ $1) \int_{0}^{1} \frac{d x}{\sqrt{1 + x} + \sqrt{1 - x} + 2}$ $2) \int_{0}^{2} \frac{l n (1 + 2 x)}{1 + x^{2}}$ $3)$ $\int_{0}^{π} \frac{x}{\sqrt{1 + \sin^{3} x}} ((3 π \cos x + 4 \sin x) \sin^{2} x + 4) d x$ $4)$ $\int_{0}^{π} \frac{x^{2} \cos^{2} x - x \sin x - \cos x - 1}{{(1 + x \sin x)}^{2}} d x .$

Question Number 54073 Answers: 1 Comments: 0

Question Number 54070 Answers: 2 Comments: 7

Evaluate : 1) ∫_(−1) ^( 1) cot^(−1) ((1/(√(1−x^2 )))).(cot^(−1) (x/(√(1−(x^2 )^(∣x∣) ))))dx 2) ∫_0 ^( (π/2)) ((sin^2 (10)θ)/(sin^2 θ)) dθ 3) ∫_0 ^( (π/4)) ((ln(cotx))/(((sinx)^(2009) +(cosx)^(2009) )^2 )).(sin2x)^(2008) dx 4) ∫_0 ^( 2) ((4x+10)/((x^2 +5x+6)^2 )) dx.

$E v a l u a t e :$ $1)$ $\int_{- 1}^{1} \cot^{- 1} (\frac{1}{\sqrt{1 - x^{2}}}) . (co t^{- 1} \frac{x}{\sqrt{1 - {(x^{2})}^{∣ x ∣}}}) d x$ $2)$ $\int_{0}^{\frac{π}{2}} \frac{\sin^{2} (10) θ}{\sin^{2} θ} d θ$ $3)$ $\int_{0}^{\frac{π}{4}} \frac{l n (c o t x)}{{({(\sin x)}^{2009} + {(\cos x)}^{2009})}^{2}} . {(\sin 2 x)}^{2008} d x$ $4)$ $\int_{0}^{2} \frac{4 x + 10}{{(x^{2} + 5 x + 6)}^{2}} d x .$

Question Number 54011 Answers: 1 Comments: 2

calculate f(a) =∫ (dx/((√(1+ax))−(√(1−ax)))) with a>0 . 2) calculate U_n =∫_(−(1/(na))) ^(1/(na)) (dx/((√(1+ax))−(√(1−ax)))) with n from N and n>1 find lim_(n→+∞) U_n and study the convergence of Σ U_n

$c a l c u l a t e f (a) = \int \frac{d x}{\sqrt{1 + a x} - \sqrt{1 - a x}} w i t h a > 0 .$ $2) c a l c u l a t e U_{n} = \int_{- \frac{1}{n a}}^{\frac{1}{n a}} \frac{d x}{\sqrt{1 + a x} - \sqrt{1 - a x}} w i t h n f r o m N a n d n > 1$ $f i n d {l i m}_{n \to + \infty} U_{n} a n d s t u d y t h e c o n v e r g e n c e o f Σ U_{n}$

Question Number 53967 Answers: 2 Comments: 1

1)calculate A_t =∫_0 ^∞ e^(−xt) sinxdx with x>0 2) by using Fubuni theorem find the value of ∫_0 ^∞ ((sinx)/x)dx .

$1) c a l c u l a t e A_{t} = \int_{0}^{\infty} e^{- x t} s i n x d x w i t h x > 0$ $2) b y u s i n g F u b u n i t h e o r e m f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{s i n x}{x} d x .$

Question Number 53966 Answers: 0 Comments: 1

let f(x) =xsinx ,2π periodic even developp f at Fourier serie .

$l e t f (x) = x s i n x, 2 π p e r i o d i c e v e n$ $d e v e l o p p f a t F o u r i e r s e r i e .$

Question Number 53956 Answers: 0 Comments: 1

give∫_0 ^1 e^(−x) ln(1−x)dx at form of serie

$g i v e \int_{0}^{1} e^{- x} l n (1 - x) d x a t f o r m o f s e r i e$

Question Number 53950 Answers: 0 Comments: 1

calculate ∫_(1/3) ^(1/2) Γ(x)Γ(1−x)dx with Γ(x) =∫_0 ^∞ t^(x−1) e^(−t) dt with x>0 .

$c a l c u l a t e \int_{\frac{1}{3}}^{\frac{1}{2}} Γ (x) Γ (1 - x) d x w i t h Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t w i t h x > 0 .$

Question Number 53958 Answers: 0 Comments: 0

find the value of ∫_0 ^1 ln(x)ln(1−x^2 )dx

$f i n d t h e v a l u e o f \int_{0}^{1} l n (x) l n (1 - x^{2}) d x$

Question Number 53963 Answers: 0 Comments: 1

let f(x) = x∣x∣ , 2π periodic odd developp f at fourier serie .

$l e t f (x) = x ∣ x ∣, 2 π p e r i o d i c o d d$ $d e v e l o p p f a t f o u r i e r s e r i e .$

Question Number 53931 Answers: 0 Comments: 1

∫x!dx

$\int x! d x$

Question Number 53843 Answers: 1 Comments: 0

Question Number 53785 Answers: 0 Comments: 1

let f(x)=∫_0 ^∞ ((tsin(tx))/(1+t^4 ))dt with x>0 1) find a explicit form of f(x) 2) find the value of ∫_0 ^∞ ((tsin(2t))/(1+t^4 ))dt.

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

Question Number 53783 Answers: 1 Comments: 0

calculate ∫_0 ^∞ (t^2 /(e^t −1))dt interms of ξ(3)

$c a l c u l a t e \int_{0}^{\infty} \frac{t^{2}}{e^{t} - 1} d t i n t e r m s o f ξ (3)$

Question Number 53782 Answers: 1 Comments: 0

calculate ∫_0 ^1 (t^2 /(1+t^3 ))dt

$c a l c u l a t e \int_{0}^{1} \frac{t^{2}}{1 + t^{3}} d t$

Question Number 53624 Answers: 0 Comments: 0

find ∫_(−∞) ^(+∞) e^(−x^2 ) (√(1+2x^2 ))dx

$f i n d \int_{- \infty}^{+ \infty} e^{- x^{2}} \sqrt{1 + 2 x^{2}} d x$

Question Number 53623 Answers: 1 Comments: 3

1) study the function f(x)=ln(x+1−(√x)) 2) determine f^(−1) (x) 3) cslculate ∫ f(x)dx snd ∫ f^(−1) (x)dx 4) dtermine ∫ f^(−1) (x^2 +f(x))dx

$1) s t u d y t h e f u n c t i o n$ $f (x) = l n (x + 1 - \sqrt{x})$ $2) d e t e r m i n e f^{- 1} (x)$ $3) c s l c u l a t e \int f (x) d x s n d$ $\int f^{- 1} (x) d x$ $4) d t e r m i n e \int f^{- 1} (x^{2} + f (x)) d x$

Question Number 53601 Answers: 1 Comments: 0

calculate ∫_0 ^∞ ((e^(−x^2 ) −e^(−x) )/x) dx .

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

Question Number 53600 Answers: 0 Comments: 1

calculate A_m =∫_0 ^∞ ((sin(mx))/(e^(2πx) −1)) dx with m>0

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

Question Number 53599 Answers: 0 Comments: 1

1) calculate A_n =∫_0 ^∞ (x^(n−1) /(e^x +1)) dx with n integr natural (n≥2) 2) find the value of ∫_0 ^∞ (x/(e^x +1))dx

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

Question Number 53483 Answers: 0 Comments: 9

Question Number 53477 Answers: 1 Comments: 1

let f(a)=∫_0 ^1 (dt/((√(x+a)) +3)) 1) calculate f(a) 2) find also ∫_0 ^1 (dt/((√(x+a))((√(x+a)) +3)^2 )) 3) find the values of integrals ∫_0 ^1 (dt/((√(x+1))+3)) and ∫_0 ^1 (dt/((√(x+1))((√(x+1))+3)^2 ))

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

Question Number 53476 Answers: 0 Comments: 0

let f(x)=∫_0 ^x t(√(2t−1))dt calculate ∣sup_(1≤x≤2) f(x) −inf_(1≤x≤2) f(x)∣

$l e t f (x) = \int_{0}^{x} t \sqrt{2 t - 1} d t c a l c u l a t e ∣ {s u p}_{1 ⩽ x ⩽ 2} f (x) - {i n f}_{1 ⩽ x ⩽ 2} f (x) ∣$

Question Number 53474 Answers: 1 Comments: 0

calculate ∫_0 ^1 ((5^(2x+1) −2^(2x−1) )/(10^x )) dx

$c a l c u l a t e \int_{0}^{1} \frac{5^{2 x + 1} - 2^{2 x - 1}}{10^{x}} d x$

Question Number 53471 Answers: 1 Comments: 3

1)find U_n = ∫_0 ^(π/4) tan^n tdt with n integr . 2) find lim_(n→+∞) U_n 3) calculate Σ_(n=0) ^∞ U_n

$1) f i n d U_{n} = \int_{0}^{\frac{π}{4}} {t a n}^{n} t d t w i t h n i n t e g r .$ $2) f i n d {l i m}_{n \to + \infty} U_{n}$ $3) c a l c u l a t e \sum_{n = 0}^{\infty} U_{n}$

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