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IntegrationQuestion and Answers: Page 278

Question Number 38651 Answers: 1 Comments: 0

If ∫_0 ^1 e^(−x^2 ) dx = a , then find the value of ∫_0 ^1 x^2 e^(−x^2 ) dx in terms of ′a′ ?

$If \int_{0}^{1} e^{- x^{2}} d x = a, then find the value$ $of \int_{0}^{1} x^{2} e^{- x^{2}} d x i n t e r m s o f^{'} a^{'} ?$

Question Number 38536 Answers: 2 Comments: 0

∫_0 ^π (dx/(√(3−cos x)))=

$\int_{0}^{π} \frac{d x}{\sqrt{3 - \cos x}} =$

Question Number 38516 Answers: 3 Comments: 1

∫_0 ^(π/2) ∣sin x − cos x∣dx

$\int_{0}^{\frac{π}{2}} ∣ \sin x - \cos x ∣ d x$

Question Number 38470 Answers: 0 Comments: 4

calculate f(t)=∫_0 ^∞ ((cos(tx))/((1+tx^2 )^2 )) dx with t≥0 2) find the values of ∫_0 ^∞ ((cos(2x))/((1+2x^2 )^2 ))dx and ∫_0 ^∞ ((cosx)/((2+x^2 )^2 ))dx

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

Question Number 38469 Answers: 0 Comments: 1

calculate f(a) = ∫_(−∞) ^(+∞) ((sin(ax))/(x^2 +x+1))dx 2) find the value of ∫_(−∞) ^(+∞) ((sin(3x))/(x^2 +x+1))dx

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

Question Number 38468 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) ((sin(2x)sh(3x))/(4+x^2 ))dx

$c a l c u l a t e \int_{- \infty}^{+ \infty} \frac{s i n (2 x) s h (3 x)}{4 + x^{2}} d x$

Question Number 38467 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) ((cos(ax)ch(bx))/(x^2 +1))dx .

$c a l c u l a t e \int_{- \infty}^{+ \infty} \frac{c o s (a x) c h (b x)}{x^{2} + 1} d x .$

Question Number 38466 Answers: 0 Comments: 1

let a from R find F_a (t)= ∫_(−∞) ^(+∞) ((cos(tx))/(a^2 +x^2 ))dx 2) calculate F_2 (3) and F_3 (2)

$l e t a f r o m R f i n d F_{a} (t) = \int_{- \infty}^{+ \infty} \frac{c o s (t x)}{a^{2} + x^{2}} d x$ $2) c a l c u l a t e F_{2} (3) a n d F_{3} (2)$

Question Number 38465 Answers: 0 Comments: 2

find f(x)= ∫_0 ^1 ln(1+xt^3 )dt with ∣x∣<1 . 2) calculate ∫_0 ^1 ln(1+4t^3 )dt and ∫_0 ^1 ln(2+t^3 )dt.

$f i n d f (x) = \int_{0}^{1} l n (1 + {x t}^{3}) d t w i t h ∣ x ∣< 1 .$ $2) c a l c u l a t e \int_{0}^{1} l n (1 + 4 t^{3}) d t a n d \int_{0}^{1} l n (2 + t^{3}) d t .$

Question Number 38464 Answers: 0 Comments: 1

let f(x)= ∫_0 ^1 ((ln(1−x^2 t^2 ))/t^2 )dt with ∣x∣<1 find f(x) at a simple form .

$l e t f (x) = \int_{0}^{1} \frac{l n (1 - x^{2} t^{2})}{t^{2}} d t w i t h ∣ x ∣< 1$ $f i n d f (x) a t a s i m p l e f o r m .$

Question Number 38463 Answers: 0 Comments: 1

calculate I = ∫_0 ^1 ((ln (1−(t^2 /4)))/t^2 )dt

$c a l c u l a t e I = \int_{0}^{1} \frac{l n (1 - \frac{t^{2}}{4})}{t^{2}} d t$

Question Number 38462 Answers: 0 Comments: 0

find ∫_1 ^(+∞) arctan(x −(1/x))dx

$f i n d \int_{1}^{+ \infty} a r c t a n (x - \frac{1}{x}) d x$

Question Number 38461 Answers: 0 Comments: 0

find f(x) = ∫_0 ^∞ arctan(1+e^(−xt) )dt with x>0 2) find ∫_0 ^∞ arctan(1+e^(−2t) )dt.

$f i n d f (x) = \int_{0}^{\infty} a r c t a n (1 + e^{- x t}) d t w i t h x > 0$ $2) f i n d \int_{0}^{\infty} a r c t a n (1 + e^{- 2 t}) d t .$

Question Number 38453 Answers: 0 Comments: 4

find f(x)=∫_0 ^∞ ((1−cos(xt))/t) e^(−xt) dt with x>0 1) find asimple form of f(x) 2) calculate ∫_0 ^∞ ((1−cos(πt))/t) e^(−t) dt 3)calculate ∫_0 ^∞ ((1−cos(3t))/t) e^(−2t) dt

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

Question Number 38451 Answers: 1 Comments: 1

new attempt to solve qu. 37630 ∫(dx/((√x)+(√(x+1))+(√(x+2))))= [t=x+1 → dx=dt] =∫(dt/((√(t−1))+(√t)+(√(t+1))))= [((to omit the roots)),(((√a)+(√b)+(√c) must be multiplied with)),(((−(√a)−(√b)+(√c))(−(√a)+(√b)−(√c))((√a)−(√b)−(√c)))),(((1/((√a)+(√b)+(√c)))=((a^(3/2) +b^(3/2) +c^(3/2) +2(√(abc))−((a+b)(√c)+(a+c)(√b)+(b+c)(√a)))/(a^2 +b^2 +c^2 −2(ab+ac+bc))))) ] =∫((t(√(t−1))+t(√t)+t(√(t+1))+2(√(t−1))−2(√(t+1))−2(√((t−1)t(t+1))))/(3t^2 −4))dt= =∫((t(√(t−1)))/(3t^2 −4))dt+∫((t(√t))/(3t^2 −4))dt+∫((t(√(t+1)))/(3t^2 −4))dt+2∫((√(t−1))/(3t^2 −4))dt−2∫((√(t+1))/(3t^2 −4))−2∫((√((t−1)t(t+1)))/(3t^2 −4))dt I think I can solve them all except the last one so please somebody try ∫((√((t−1)t(t+1)))/(3t^2 −4))dt=? I will do the others tomorrow

$new attempt to solve qu . 37630$ $\int \frac{d x}{\sqrt{x} + \sqrt{x + 1} + \sqrt{x + 2}} =$ $[t = x + 1 \to d x = d t]$ $= \int \frac{d t}{\sqrt{t - 1} + \sqrt{t} + \sqrt{t + 1}} =$ $[\begin{matrix} to omit the roots \\ \sqrt{a} + \sqrt{b} + \sqrt{c} must be multiplied with \\ (- \sqrt{a} - \sqrt{b} + \sqrt{c}) (- \sqrt{a} + \sqrt{b} - \sqrt{c}) (\sqrt{a} - \sqrt{b} - \sqrt{c}) \\ \frac{1}{\sqrt{a} + \sqrt{b} + \sqrt{c}} = \frac{a^{3 / 2} + b^{3 / 2} + c^{3 / 2} + 2 \sqrt{a b c} - ((a + b) \sqrt{c} + (a + c) \sqrt{b} + (b + c) \sqrt{a})}{a^{2} + b^{2} + c^{2} - 2 (a b + a c + b c)} \end{matrix}]$ $= \int \frac{t \sqrt{t - 1} + t \sqrt{t} + t \sqrt{t + 1} + 2 \sqrt{t - 1} - 2 \sqrt{t + 1} - 2 \sqrt{(t - 1) t (t + 1)}}{3 t^{2} - 4} d t =$ $= \int \frac{t \sqrt{t - 1}}{3 t^{2} - 4} d t + \int \frac{t \sqrt{t}}{3 t^{2} - 4} d t + \int \frac{t \sqrt{t + 1}}{3 t^{2} - 4} d t + 2 \int \frac{\sqrt{t - 1}}{3 t^{2} - 4} d t - 2 \int \frac{\sqrt{t + 1}}{3 t^{2} - 4} - 2 \int \frac{\sqrt{(t - 1) t (t + 1)}}{3 t^{2} - 4} d t$ $I think I can solve them all except the last one$ $so please somebody try$ $\int \frac{\sqrt{(t - 1) t (t + 1)}}{3 t^{2} - 4} d t = ?$ $I will do the others tomorrow$

Question Number 38460 Answers: 0 Comments: 1

let ∣x∣>1 find the value of F(x)=∫_0 ^∞ ln(1+xt^2 )dt 2)calculate ∫_0 ^∞ ln(1+3t^2 )dt .

$l e t ∣ x ∣> 1 f i n d t h e v a l u e o f$ $F (x) = \int_{0}^{\infty} l n (1 + {x t}^{2}) d t$ $2) c a l c u l a t e \int_{0}^{\infty} l n (1 + 3 t^{2}) d t .$

Question Number 38458 Answers: 0 Comments: 3

let ∣x∣<1 calculate F(x)=∫_0 ^1 ln(1+xt^2 )dt 2) find the value of ∫_0 ^1 ln(1 +(1/2)t^2 )dt 3)find the value of A(θ) =∫_0 ^1 ln(1+sinθ t^2 )dt .

$l e t ∣ x ∣< 1 c a l c u l a t e F (x) = \int_{0}^{1} l n (1 + {x t}^{2}) d t$ $2) f i n d t h e v a l u e o f \int_{0}^{1} l n (1 + \frac{1}{2} t^{2}) d t$ $3) f i n d t h e v a l u e o f A (θ) = \int_{0}^{1} l n (1 + s i n θ t^{2}) d t .$

Question Number 38457 Answers: 0 Comments: 0

f is a C^2 function prove that 1)L(f^′ )=x L(f)−f(0^+ ) 2)L(f^(′′) )=x^2 L(f) −xf(0^+ )−f^′ (0^+ ) L means Laplace transform.

$f i s a C^{2} f u n c t i o n p r o v e t h a t$ $1) L (f^{^{'}}) = x L (f) - f (0^{+})$ $2) L (f^{^{″}}) = x^{2} L (f) - x f (0^{+}) - f^{^{'}} (0^{+})$ $L m e a n s L a p l a c e t r a n s f o r m .$

Question Number 38454 Answers: 0 Comments: 3

let f(x)=∫_0 ^∞ ((1−cos(xt^2 ))/t^2 ) e^(−xt^2 ) dt with x>0 1) find a simple form of f(x) 2) calculate ∫_0 ^∞ ((1−cos(2t^2 ))/t^2 ) e^(−3t^2 ) dt .

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

Question Number 38405 Answers: 1 Comments: 2

If f(3)=3;f(1)=2 ⇒∫_1 ^3 f(x)f^′ (x)dx=.....

$If f (3) = 3; f (1) = 2$ $\Rightarrow \int_{1}^{3} f (x) f^{^{'}} (x) dx = . . . . .$

Question Number 38397 Answers: 2 Comments: 2

evaluate ∫secxdx

$e v a l u a t e \int s e c x d x$

Question Number 38384 Answers: 0 Comments: 1

∫x^(−3cosx) dx please is this possible? If possibld then prove it. Thanks in advance.

$\int x^{- 3 c o s x} d x$ $p l e a s e i s t h i s p o s s i b l e ?$ $I f p o s s i b l d t h e n p r o v e i t .$ $T h a n k s i n a d v a n c e .$

Question Number 38310 Answers: 1 Comments: 4

let f(x)=∫_0 ^(+∞) ((arctan(xt))/(1+t^2 ))dt with x≥0 1) calculate f^′ (x) then a simple form of f(x) 2) calculate ∫_0 ^(+∞) ((arctant)/(1+t^2 ))dt 3) calculate ∫_0 ^(+∞) ((arctan(2t))/(1+t^2 ))dt

$l e t f (x) = \int_{0}^{+ \infty} \frac{a r c t a n (x t)}{1 + t^{2}} d t w i t h x ⩾ 0$ $1) c a l c u l a t e f^{^{'}} (x) t h e n a s i m p l e f o r m o f f (x)$ $2) c a l c u l a t e \int_{0}^{+ \infty} \frac{a r c t a n t}{1 + t^{2}} d t$ $3) c a l c u l a t e \int_{0}^{+ \infty} \frac{a r c t a n (2 t)}{1 + t^{2}} d t$

Question Number 38211 Answers: 0 Comments: 2

let x>0 and F(x)= ∫_0 ^(+∞) ((arctan(xt^2 ))/(1+t^2 ))dt 1) find a simple form of F(x) 2)find the value of ∫_0 ^∞ ((arctan(2t^2 ))/(1+t^2 ))dt 3)find the value of ∫_0 ^∞ ((arctan(3t^2 ))/(1+t^2 ))dt.

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

Question Number 38210 Answers: 2 Comments: 4

let f(a)= ∫_0 ^π (dθ/(a +sin^2 θ)) (a from R) 1) find f(a) 2)calculate g(a)= ∫_0 ^π (dθ/((a+sin^2 θ)^2 )) 3)calculate ∫_0 ^π (dθ/(1+sin^2 θ)) and ∫_0 ^π (dθ/(2+sin^2 θ)) 4) calculate ∫_0 ^π (dθ/((3 +sin^2 θ)^2 )) .

$l e t f (a) = \int_{0}^{π} \frac{d θ}{a + {s i n}^{2} θ} (a f r o m R)$ $1) f i n d f (a)$ $2) c a l c u l a t e g (a) = \int_{0}^{π} \frac{d θ}{{(a + {s i n}^{2} θ)}^{2}}$ $3) c a l c u l a t e \int_{0}^{π} \frac{d θ}{1 + {s i n}^{2} θ} a n d \int_{0}^{π} \frac{d θ}{2 + {s i n}^{2} θ}$ $4) c a l c u l a t e \int_{0}^{π} \frac{d θ}{{(3 + {s i n}^{2} θ)}^{2}} .$

Question Number 38209 Answers: 0 Comments: 2

let f(x)=e^(−x) cosx developp f at fourier serie 1) find the value of Σ_(n=−∞) ^(+∞) (((−1)^n )/(1+n^2 )) 2) calculate Σ_(n=0) ^∞ (1/(n^2 +1)) .

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

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