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IntegrationQuestion and Answers: Page 314

Question Number 27615 Answers: 0 Comments: 2

∫x^(5/2) (1−x)^(3/2) dx

$\int x^{5 / 2} {(1 - x)}^{3 / 2} d x$

Question Number 27614 Answers: 0 Comments: 2

∫((cosx)/(2−cosx))dx

$\int \frac{c o s x}{2 - c o s x} d x$

Question Number 27613 Answers: 1 Comments: 1

find the value of ∫_0 ^∞ e^(−[x] −x) dx .

$f i n d t h e v a l u e o f \int_{0}^{\infty} e^{- [x] - x} d x .$

Question Number 27612 Answers: 1 Comments: 0

∫(1/(3+cos^2 x))dx

$\int \frac{1}{3 + {c o s}^{2} x} d x$

Question Number 27611 Answers: 1 Comments: 0

∫(1/(2sin^2 x + 4cos^2 x))dx

$\int \frac{1}{2 \sin^{2} x + 4 \cos^{2} x} d x$

Question Number 27607 Answers: 0 Comments: 0

Question Number 27600 Answers: 0 Comments: 1

find ∫_0 ^π (t/(2+sint)) dt

$f i n d \int_{0}^{π} \frac{t}{2 + s i n t} d t$

Question Number 27597 Answers: 0 Comments: 0

find ∫ ((√(cos(2x)))/(cosx)) dx.

$f i n d \int \frac{\sqrt{c o s (2 x)}}{c o s x} d x .$

Question Number 27596 Answers: 0 Comments: 1

find ∫ ^3 (√( x^2 −x^3 )) dx

$f i n d \int^{3} \sqrt{x^{2} - x^{3}} d x$

Question Number 27595 Answers: 0 Comments: 1

find ∫∫_D xy(√( x^2 +y^2 )) dxdy with D={ (x,y)∈R^2 / x^2 +2y^2 ≤1 ,x≥0 ,y ≥0}

$f i n d \int \int_{D} x y \sqrt{x^{2} + y^{2}} d x d y w i t h$ $D = {(x, y) \in R^{2} / x^{2} + 2 y^{2} ⩽ 1, x ⩾ 0, y ⩾ 0}$

Question Number 27539 Answers: 2 Comments: 1

Question Number 27502 Answers: 0 Comments: 1

find ∫_0 ^(π/2) ((ln(1+xsin^2 t))/(sin^2 t))dt with −1<x<1 .

$f i n d \int_{0}^{\frac{π}{2}} \frac{l n (1 + {x s i n}^{2} t)}{{s i n}^{2} t} d t w i t h - 1 < x < 1 .$

Question Number 27500 Answers: 0 Comments: 2

find ∫∫_Δ (√(4 −x^2 −y^2 )) dxdy with Δ={(x,y) ∈R^2 / x^2 +y^2 ≤2x}

$f i n d \int \int_{Δ} \sqrt{4 - x^{2} - y^{2}} d x d y w i t h$ $Δ = {(x, y) \in R^{2} / x^{2} + y^{2} ⩽ 2 x}$

Question Number 27496 Answers: 0 Comments: 0

let give f(x)= ∫_0 ^∝ (1/(√t)) e^(−(1+ix)t) dt calculate f^′ (x) prove that ∃λ∈R/(x+i)^2 (f(x))^2 = λ then find ∫_0 ^∝ e^(−t^2 ) dt .

$l e t g i v e f (x) = \int_{0}^{\propto} \frac{1}{\sqrt{t}} e^{- (1 + i x) t} d t$ $c a l c u l a t e f^{^{'}} (x) p r o v e t h a t \exists λ \in R / {(x + i)}^{2} {(f (x))}^{2} = λ$ $t h e n f i n d \int_{0}^{\propto} e^{- t^{2}} d t .$

Question Number 27495 Answers: 0 Comments: 1

find α and β from R /∫_0 ^π (αt^2 +βt)cos(nt)dt= (1/n^2 ) for all number n from N^(∗ ) then find Σ_(n=1) ^∝ (1/n^2 ) .

$f i n d α a n d β f r o m R / \int_{0}^{π} (α t^{2} + β t) c o s (n t) d t = \frac{1}{n^{2}}$ $f o r a l l n u m b e r n f r o m N^{*} t h e n f i n d$ $\sum_{n = 1}^{\propto} \frac{1}{n^{2}} .$

Question Number 27481 Answers: 0 Comments: 1

find the value of ∫_0 ^∝ ((√x)/(e^x −1))dx .

$f i n d t h e v a l u e o f \int_{0}^{\propto} \frac{\sqrt{x}}{e^{x} - 1} d x .$

Question Number 27464 Answers: 0 Comments: 5

Show that: ∫_( 0) ^( 2π) ((cos(3x))/(5 − 4cos(x))) dx = (π/(12))

$Show that : \int_{0}^{2 π} \frac{\cos (3 x)}{5 - 4 \cos (x)} dx = \frac{π}{12}$

Question Number 27410 Answers: 0 Comments: 2

∫_0 ^∞ ((sinxdx)/x)

$\underset{0}{\int^{\infty}} \frac{s i n x d x}{x}$

Question Number 27400 Answers: 1 Comments: 4

Evaluate ∫_r ^0 (√(x/(r−x))) dx

$E v a l u a t e \underset{r}{\int^{0}} \sqrt{\frac{x}{r - x}} d x$

Question Number 27392 Answers: 0 Comments: 0

Show that the integral: ∫ e^(−x^2 ) dx Can′t be calculated trivially.

$Show that the integral :$ $\int e^{- x^{2}} dx$ ${Can}^{'} t be calculated trivially .$

Question Number 27345 Answers: 0 Comments: 1

prove that ∫_0 ^∞ (t^(x−1) /(e^t −1))dt =ξ(x)Γ(x) with ξ(x)= Σ_(n=1) ^∝ (1/n^x ) and Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt ( x>1)

$p r o v e t h a t \int_{0}^{\infty} \frac{t^{x - 1}}{e^{t} - 1} d t = ξ (x) Γ (x)$ $w i t h ξ (x) = \sum_{n = 1}^{\propto} \frac{1}{n^{x}} a n d Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t$ $(x > 1)$

Question Number 27342 Answers: 0 Comments: 1

find f(x)= ∫_0 ^∞ (e^(−x(1+t^2 )) /(1+t^2 )) dt interms ofx with x≥0 and calculate ∫_0 ^∞ e^(−t^2 ) dt .

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

Question Number 27341 Answers: 0 Comments: 0

prove that ∫_0 ^∞ e^(−(t^2 +(1/t^2 ))) dt is convergeny and find its value .

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

Question Number 27309 Answers: 1 Comments: 0

find the value of ∫_0 ^∝ (((−1)^([x]) )/((2x+1)^2 ))dx

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

Question Number 27282 Answers: 0 Comments: 1

∫log(2+x^2 )dx

$\int l o g (2 + x^{2}) d x$

Question Number 27271 Answers: 1 Comments: 0

Proof ∫(1/(a^2 −x^2 ))dx =(1/(2a))ln∣((a+x)/(a−x))∣+c

$P r o o f$ $\int \frac{1}{a^{2} - x^{2}} d x = \frac{1}{2 a} l n ∣ \frac{a + x}{a - x} ∣ + c$

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