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IntegrationQuestion and Answers: Page 291

Question Number 35048 Answers: 0 Comments: 0

find ∫ (dx/(cos(sinx)))

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

Question Number 35046 Answers: 0 Comments: 0

find F(x)= ∫_0 ^π ln( 1+x sin^2 t)dt with ∣x∣<1 2) calculate ∫_0 ^π ln(1+(1/2)sin^2 t)dt

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

Question Number 35045 Answers: 0 Comments: 0

find f(x)=∫_0 ^∞ ((arctan(xt))/(1+t^2 ))dt .

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

Question Number 35044 Answers: 1 Comments: 1

1)find ∫ (√(1+t^2 )) dt 2) calculate ∫_1 ^(√3) (√(1+t^2 )) dt

$1) f i n d \int \sqrt{1 + t^{2}} d t$ $2) c a l c u l a t e \int_{1}^{\sqrt{3}} \sqrt{1 + t^{2}} d t$

Question Number 35043 Answers: 1 Comments: 0

let t>0 and F(t) =∫_0 ^∞ ((sin(x^2 ) e^(−tx^2 ) )/x^2 )dx calculate (dF/dt)(t).

$l e t t > 0 a n d F (t) = \int_{0}^{\infty} \frac{s i n (x^{2}) e^{- {t x}^{2}}}{x^{2}} d x$ $c a l c u l a t e \frac{d F}{d t} (t) .$

Question Number 35018 Answers: 1 Comments: 0

∫∫∫((dxdydz)/((x+y+z+1)^3 )) bounded by the coordinate planes and the plane x+y+z=1 .

$\int \int \int \frac{d x d y d z}{{(x + y + z + 1)}^{3}} b o u n d e d b y t h e$ $c o o r d i n a t e p l a n e s a n d t h e p l a n e$ $x + y + z = 1 .$

Question Number 35015 Answers: 2 Comments: 0

∫(x^2 /((1+x^3 )^2 ))dx

$\int \frac{x^{2}}{{(1 + x^{3})}^{2}} d x$

Question Number 34992 Answers: 1 Comments: 1

∫_0 ^π ((cos(x))/(1+2sin(2x)))dx

$\underset{0}{\int^{π}} \frac{\cos (x)}{1 + 2 \sin (2 x)} d x$

Question Number 34956 Answers: 3 Comments: 2

Question Number 34911 Answers: 1 Comments: 1

find ∫_2 ^3 ((2x^2 +3)/((x−1)^2 (x^2 +1))) dx

$f i n d \int_{2}^{3} \frac{2 x^{2} + 3}{{(x - 1)}^{2} (x^{2} + 1)} d x$

Question Number 34910 Answers: 0 Comments: 1

find J_(n,p) =∫_0 ^∞ x^n e^(−(x^2 /p)) dx with p>0 and n integr

$f i n d J_{n, p} = \int_{0}^{\infty} x^{n} e^{- \frac{x^{2}}{p}} d x w i t h p > 0 a n d n i n t e g r$

Question Number 34901 Answers: 0 Comments: 3

∫_(−π/2) ^(+π/2) (√(cos^(2n−1) x−cos^(2n+1) x))dx =[−((2cos^((2n+1)/2) x)/(2n+1))]_(−π/2) ^(+π/2) =0? What is the mistake in above? ∫_(−π/2) ^(+π/2) (√(cos^(2n−1) x−cos^(2n+1) x))dx =2∫_0 ^(π/2) (√(cos^(2n−1) x−cos^(2n+1) x))dx =(4/(2n+1)) (this is correct answer)

$\int_{- π / 2}^{+ π / 2} \sqrt{\cos^{2 n - 1} x - \cos^{2 n + 1} x} d x$ $= {[- \frac{{2 \cos}^{\frac{2 n + 1}{2}} x}{2 n + 1}]}_{- π / 2}^{+ π / 2} = 0 ?$ $What is the mistake in above ?$ $\int_{- π / 2}^{+ π / 2} \sqrt{\cos^{2 n - 1} x - \cos^{2 n + 1} x} d x$ $= 2 \int_{0}^{π / 2} \sqrt{\cos^{2 n - 1} x - \cos^{2 n + 1} x} d x$ $= \frac{4}{2 n + 1} (this is correct answer)$

Question Number 34866 Answers: 0 Comments: 0

find f(x)=∫_0 ^∞ ((arctan(x(t +(1/t))))/(1+t^2 ))dt

$f i n d f (x) = \int_{0}^{\infty} \frac{a r c t a n (x (t + \frac{1}{t}))}{1 + t^{2}} d t$

Question Number 34862 Answers: 2 Comments: 8

find the value of f(x) = ∫_0 ^π ((cosx)/(1+2sin(2x)))dx

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

Question Number 34827 Answers: 1 Comments: 5

Find ∫ Sin^6 x dx

$F i n d \int {S i n}^{6} x d x$

Question Number 34771 Answers: 0 Comments: 1

let A(x)= ∫_0 ^1 ln(1+ix^2 )dx find a simple form of f(x) (x∈R)

$l e t A (x) = \int_{0}^{1} l n (1 + {i x}^{2}) d x$ $f i n d a s i m p l e f o r m o f f (x) (x \in R)$

Question Number 34720 Answers: 0 Comments: 0

let B(p,q) = ∫_0 ^1 x^(p−1) (1−x)^(q−1) dx calculate B((1/3), (1/3)) 2) calculate B((1/2) ,(2/3)) .

$l e t B (p, q) = \int_{0}^{1} x^{p - 1} {(1 - x)}^{q - 1} d x$ $c a l c u l a t e B (\frac{1}{3}, \frac{1}{3})$ $2) c a l c u l a t e B (\frac{1}{2}, \frac{2}{3}) .$

Question Number 34717 Answers: 0 Comments: 1

let I_n = ∫∫_([(1/n),n]^2 ) (((√(xy)) dxdy)/(2 +x^2 +y^2 )) find lim I_n when n→+∞.

$l e t I_{n} = \int \int_{{[\frac{1}{n}, n]}^{2}} \frac{\sqrt{x y} d x d y}{2 + x^{2} + y^{2}}$ $f i n d l i m I_{n} w h e n n \to + \infty .$

Question Number 34716 Answers: 0 Comments: 1

calculate ∫∫_w x(√(x^2 +y^2 )) dxdy w ={(x,y)/ x^2 +y^2 ≤3 }

$c a l c u l a t e \int \int_{w} x \sqrt{x^{2} + y^{2}} d x d y$ $w = {(x, y) / x^{2} + y^{2} ⩽ 3}$

Question Number 34715 Answers: 0 Comments: 0

calculate ∫∫_(0≤x≤y≤1) ((dxdy)/((x^2 +1)(y^2 +3))) .

$c a l c u l a t e \int \int_{0 ⩽ x ⩽ y ⩽ 1} \frac{d x d y}{(x^{2} + 1) (y^{2} + 3)} .$

Question Number 34714 Answers: 0 Comments: 1

calculate ∫∫_(x^2 +2y^2 ≤1) (x^2 −y^2 )dxdy

$c a l c u l a t e \int \int_{x^{2} + 2 y^{2} ⩽ 1} (x^{2} - y^{2}) d x d y$

Question Number 34713 Answers: 0 Comments: 1

let a>0 calculate ∫∫_(x^2 +y^2 ≤3) (1/(2 +x^2 +y^2 ))dxdy.

$l e t a > 0 c a l c u l a t e \int \int_{x^{2} + y^{2} ⩽ 3} \frac{1}{2 + x^{2} + y^{2}} d x d y .$

Question Number 34675 Answers: 0 Comments: 0

provethat e = Σ_(k=0) ^n (1/(k!)) +∫_0 ^1 (((1−t)^n )/(n!)) e^t dt .

$p r o v e t h a t e = \sum_{k = 0}^{n} \frac{1}{k!} + \int_{0}^{1} \frac{{(1 - t)}^{n}}{n!} e^{t} d t .$

Question Number 34674 Answers: 0 Comments: 0

find ∫_0 ^π ((x sinx)/(1+cos^2 x)) dx

$f i n d \int_{0}^{π} \frac{x s i n x}{1 + {c o s}^{2} x} d x$

Question Number 34662 Answers: 0 Comments: 0

calculate I(a) =∫_(1/a) ^a ((ln(x))/(1+x^2 )) dx with a>0 2) calculate ∫_0 ^(+∞) ((ln(x))/(1+x^2 )) dx .

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

Question Number 34661 Answers: 0 Comments: 0

let f(x)= ∫_0 ^1 (e^(−(1+t^2 )x) /(1+t^2 )) dt find a simple form of f(x)

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

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