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IntegrationQuestion and Answers: Page 252

Question Number 48494 Answers: 1 Comments: 0

find A_n =∫_0 ^(π/2) ((1−cos(n+1)x)/(2sin((x/2))))dx .

$f i n d A_{n} = \int_{0}^{\frac{π}{2}} \frac{1 - c o s (n + 1) x}{2 s i n (\frac{x}{2})} d x .$

Question Number 48491 Answers: 0 Comments: 0

prove that ∫_0 ^∞ (((1+t)^(−(3/4)) −(1+t)^(−(1/4)) )/t)dt is convergent and find its value .

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

Question Number 48484 Answers: 1 Comments: 0

Question Number 48289 Answers: 0 Comments: 4

Evaluate ∫_0 ^1 ((Log(x))/(x^2 +2x+3)) dx

$E v a l u a t e \underset{0}{\int^{1}} \frac{L o g (x)}{x^{2} + 2 x + 3} d x$

Question Number 48264 Answers: 0 Comments: 0

let f(x)=∫_0 ^(2π) ((sin(2t))/(1+x cos(t)))dt 1) find a explicit form of f(x) 2) find also g(x)=∫_0 ^(2π) ((sin(2t)cost)/((1+xcost)^2 ))dt 3)find the value of ∫_0 ^(2π) ((sin(2t))/(1+3 cos(t)))dt and ∫_0 ^(2π) ((cost sin(2t))/((1+3cost)^2 ))dt .

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

Question Number 48261 Answers: 0 Comments: 4

let f(x) =∫_(1/2) ^1 (dt/(2+ch(xt))) 1) find a explicit form of f(x) 2) calculate g(x)=∫_(1/2) ^1 ((tsh(xt))/((2+ch(xt))^2 ))dt 3) find the value of ∫_(1/2) ^1 (dt/(2+ch(3t))) and ∫_(1/2) ^1 ((tsh(2t))/((2+ch(2t))^2 ))dt 4) let u_n =∫_(1/2) ^1 (dt/(2+ch(nt))) study the convergence of Σu_n and Σ(u_n /n) .

$l e t f (x) = \int_{\frac{1}{2}}^{1} \frac{d t}{2 + c h (x t)}$ $1) f i n d a e x p l i c i t f o r m o f f (x)$ $2) c a l c u l a t e g (x) = \int_{\frac{1}{2}}^{1} \frac{t s h (x t)}{{(2 + c h (x t))}^{2}} d t$ $3) f i n d t h e v a l u e o f \int_{\frac{1}{2}}^{1} \frac{d t}{2 + c h (3 t)} a n d \int_{\frac{1}{2}}^{1} \frac{t s h (2 t)}{{(2 + c h (2 t))}^{2}} d t$ $4) l e t u_{n} = \int_{\frac{1}{2}}^{1} \frac{d t}{2 + c h (n t)} s t u d y t h e c o n v e r g e n c e o f Σ u_{n}$ $a n d Σ \frac{u_{n}}{n} .$

Question Number 48255 Answers: 0 Comments: 1

calculate A_λ =∫_0 ^∞ ((cos(λsinx)−sin(λcosx))/(x^2 +λ^2 ))dx λ from R.

$c a l c u l a t e A_{λ} = \int_{0}^{\infty} \frac{c o s (λ s i n x) - s i n (λ c o s x)}{x^{2} + λ^{2}} d x$ $λ f r o m R .$

Question Number 48239 Answers: 1 Comments: 1

q.....∫(dx/(sin x cos x+2cos^2 x)), please solve

$q . . . . . \int \frac{d x}{\sin x \cos x + 2 \cos^{2} x}, p l e a s e s o l v e$

Question Number 48182 Answers: 1 Comments: 0

Question Number 48178 Answers: 1 Comments: 1

find ∫ ((sin(πx))/(3 +cos(2πx)))dx

$f i n d \int \frac{s i n (π x)}{3 + c o s (2 π x)} d x$

Question Number 48177 Answers: 0 Comments: 2

find lim_(x→0) ∫_(x+1) ^(2x+1) ((tarctan(t^2 +1))/(1+(1+t^2 )^2 ))dt

$f i n d {l i m}_{x \to 0} \int_{x + 1}^{2 x + 1} \frac{t a r c t a n (t^{2} + 1)}{1 + {(1 + t^{2})}^{2}} d t$

Question Number 48175 Answers: 1 Comments: 2

calculate lim_(x→0) ∫_x ^x^2 ((ln(1+t))/(sin(t)))dt

$c a l c u l a t e {l i m}_{x \to 0} \int_{x}^{x^{2}} \frac{l n (1 + t)}{s i n (t)} d t$

Question Number 48173 Answers: 1 Comments: 2

calculate ∫ ((arctan(x))/(√(1+x^2 )))dx

$c a l c u l a t e \int \frac{a r c t a n (x)}{\sqrt{1 + x^{2}}} d x$

Question Number 48172 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((sin(2cos(x^2 +1)))/(1+x^2 ))dx

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

Question Number 48171 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((sin(cosx))/(x^2 +3))dx

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

Question Number 48170 Answers: 1 Comments: 1

calculate ∫_0 ^∞ ((cos(sin(x^2 )))/(1+2x^2 ))dx

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

Question Number 48127 Answers: 1 Comments: 0

Question Number 48104 Answers: 1 Comments: 0

solve this ∫(2 sinx+cosx)/(2+3sinx+sin^(2x) ) dx

$solve this$ $\int (2 sinx + cosx) / (2 + 3 sinx + \sin^{2 x}) dx$

Question Number 48078 Answers: 1 Comments: 0

Question Number 48067 Answers: 0 Comments: 1

let y>0 give ∫_0 ^∞ (x^y /(e^x −1))dx at form of series.

$l e t y > 0 g i v e \int_{0}^{\infty} \frac{x^{y}}{e^{x} - 1} d x a t f o r m o f s e r i e s .$

Question Number 48064 Answers: 1 Comments: 1

calculate A =∫_0 ^1 (1+x^2 )(√(1−x^2 ))dx −∫_0 ^1 (1−x^2 )(√(1+x^2 ))dx

$c a l c u l a t e A = \int_{0}^{1} (1 + x^{2}) \sqrt{1 - x^{2}} d x - \int_{0}^{1} (1 - x^{2}) \sqrt{1 + x^{2}} d x$

Question Number 48063 Answers: 0 Comments: 0

let W(x) =∫_(−∞) ^(+∞) ((arctan(xt^2 ))/(2+t^2 ))dt 1) find a explicit form of f(x) 2) find the value of ∫_(−∞) ^(+∞) (t^2 /((2+t^2 )(1+x^2 t^4 )))dt .

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

Question Number 48062 Answers: 0 Comments: 0

calculate ∫_(−∞) ^(+∞) (((x^2 −3)sin(2x^2 ))/((x^2 +1)^3 ))dx

$c a l c u l a t e \int_{- \infty}^{+ \infty} \frac{(x^{2} - 3) s i n (2 x^{2})}{{(x^{2} + 1)}^{3}} d x$

Question Number 48057 Answers: 2 Comments: 1

Question Number 48043 Answers: 0 Comments: 1

let f(α) =∫_(−∞) ^(+∞) ((cos(αx^3 ))/(x^2 +8)) dx 1)calculate f(α) 2) calculate ∫_(−∞) ^(+∞) ((cos(2x^3 ))/(x^2 +8))dx .

$l e t f (α) = \int_{- \infty}^{+ \infty} \frac{c o s (α x^{3})}{x^{2} + 8} d x$ $1) c a l c u l a t e f (α)$ $2) c a l c u l a t e \int_{- \infty}^{+ \infty} \frac{c o s (2 x^{3})}{x^{2} + 8} d x .$

Question Number 48042 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) ((2x+1)/((x^2 +i)(x^2 +4)))dx (i^2 =−1)

$f i n d t h e v a l u e o f \int_{- \infty}^{+ \infty} \frac{2 x + 1}{(x^{2} + i) (x^{2} + 4)} d x (i^{2} = - 1)$

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