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IntegrationQuestion and Answers: Page 213

Question Number 67527 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) ((1+x^2 )/(1+x^4 ))dx

$c a l c u l a t e \int_{- \infty}^{+ \infty} \frac{1 + x^{2}}{1 + x^{4}} d x$

Question Number 67526 Answers: 0 Comments: 1

find the value of ∫_0 ^(2π) (dx/(3+2sinx +cosx))

$f i n d t h e v a l u e o f \int_{0}^{2 π} \frac{d x}{3 + 2 s i n x + c o s x}$

Question Number 67525 Answers: 0 Comments: 3

let a>b>0 calculate ∫_0 ^(2π) (dx/((a+bsinx)^2 ))

$l e t a > b > 0 c a l c u l a t e \int_{0}^{2 π} \frac{d x}{{(a + b s i n x)}^{2}}$

Question Number 67517 Answers: 0 Comments: 0

let z ∈C and ∣z∣<1 prove that (z/(1−z^2 )) +(z^2 /(1−z^4 )) +.....+(z^2^n /(1−z^2^(n+1) ))+...=(z/(1−z)) (z/(1+z)) +((2z^2 )/(1+z^2 )) +....+((2^n z^2^n )/(1+z^2^n )) +....=(z/(1−z))

$l e t z \in C a n d ∣ z ∣< 1 p r o v e t h a t$ $\frac{z}{1 - z^{2}} + \frac{z^{2}}{1 - z^{4}} + . . . . . + \frac{z^{2^{n}}}{1 - z^{2^{n + 1}}} + . . . = \frac{z}{1 - z}$ $\frac{z}{1 + z} + \frac{2 z^{2}}{1 + z^{2}} + . . . . + \frac{2^{n} z^{2^{n}}}{1 + z^{2^{n}}} + . . . . = \frac{z}{1 - z}$

Question Number 67513 Answers: 0 Comments: 0

∫x^(n ) lnx/n^x dx

$\int x^{n} l n x / n^{x} d x$

Question Number 67467 Answers: 0 Comments: 0

Find f(x)=∫_0 ^∞ (( tlnt)/((1+t^2 )^x )) dt

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

Question Number 67466 Answers: 0 Comments: 0

let consider for all n≥1 the real (t)_n =t(t+1).....(t+n−1) Find L_n = ∫_0 ^∞ (((t)_1 )/((t)_(n+1) )) dt

$l e t c o n s i d e r f o r a l l n ⩾ 1 t h e r e a l {(t)}_{n} = t (t + 1) . . . . . (t + n - 1)$ $F i n d L_{n} = \int_{0}^{\infty} \frac{{(t)}_{1}}{{(t)}_{n + 1}} d t$

Question Number 67463 Answers: 1 Comments: 3

Find Find K=∫_0 ^(π/2) (√(tanθ)) dθ

$F i n d$ $F i n d K = \int_{0}^{\frac{π}{2}} \sqrt{t a n θ} d θ$

Question Number 67462 Answers: 0 Comments: 2

Calculate when a,b are positive reals f(a,b)= ∫_0 ^1 ((t^a −t^b )/(lnt)) dt

$C a l c u l a t e w h e n a, b a r e p o s i t i v e r e a l s f (a, b) = \int_{0}^{1} \frac{t^{a} - t^{b}}{l n t} d t$

Question Number 68043 Answers: 1 Comments: 1

∫_(π/2) ^π e^(cosx) (√(1−e^(cosx) )) sinx dx

$\int_{π / 2}^{π} e^{c o s x} \sqrt{1 - e^{c o s x}} s i n x d x$

Question Number 67392 Answers: 0 Comments: 0

∫(6x

$\int (6 x$

Question Number 67385 Answers: 0 Comments: 0

find ∫ x((√((1−x^2 )/(1+x^2 ))))dx

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

Question Number 67374 Answers: 0 Comments: 3

find ∫ (1+(1/x^2 ))arctan(1−(1/x))dx

$f i n d \int (1 + \frac{1}{x^{2}}) a r c t a n (1 - \frac{1}{x}) d x$

Question Number 67373 Answers: 1 Comments: 4

simplify S_n (x) =Σ_(k=0) ^n C_n ^k cos^4 (πkx) 2) calculate I_n =∫_0 ^(1/3) S_n (x)dx

$s i m p l i f y S_{n} (x) = \sum_{k = 0}^{n} C_{n}^{k} {c o s}^{4} (π k x)$ $2) c a l c u l a t e I_{n} = \int_{0}^{\frac{1}{3}} S_{n} (x) d x$

Question Number 67359 Answers: 1 Comments: 1

∫siny/y dy

$\int s i n y / y d y$

Question Number 67342 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^∞ ((sin(2x^2 ))/((x^2 −x +3)^3 ))dx

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

Question Number 67310 Answers: 1 Comments: 2

calculate ∫_(−∞) ^(+∞) (dx/(x^4 +x^2 +1))

$c a l c u l a t e \int_{- \infty}^{+ \infty} \frac{d x}{x^{4} + x^{2} + 1}$

Question Number 67246 Answers: 2 Comments: 4

Integrate: 1) ∫_3 ^( ∞) ((1/x dx)/(ln(x)(√(ln^2 x−1)))) 2) ∫_1 ^∞ ((e^x dx)/(1+e^(2x) )) 3) ∫_1 ^∞ ((2^x dx)/(x+1)) 4) ∫_2 ^∞ ((√x)/(ln(x)))dx

$I n t e g r a t e :$ $1) \underset{3}{\int^{\infty}} \frac{1 / x d x}{l n (x) \sqrt{{l n}^{2} x - 1}}$ $2) \underset{1}{\int^{\infty}} \frac{e^{x} d x}{1 + e^{2 x}}$ $3) \underset{1}{\int^{\infty}} \frac{2^{x} d x}{x + 1}$ $4) \underset{2}{\int^{\infty}} \frac{\sqrt{x}}{l n (x)} d x$

Question Number 67235 Answers: 0 Comments: 1

find ∫_(−(π/3)) ^(π/3) x^2 {cosx−sinx}^3 dx

$f i n d \int_{- \frac{π}{3}}^{\frac{π}{3}} x^{2} {c o s x - s i n x}^{3} d x$

Question Number 67233 Answers: 0 Comments: 1

calculate ∫_0 ^1 ((xdx)/(√(1+x^4 )))

$c a l c u l a t e \int_{0}^{1} \frac{x d x}{\sqrt{1 + x^{4}}}$

Question Number 67231 Answers: 2 Comments: 0

find ∫x/x^5 −1) dx

$f i n d \int x / x^{5} - 1) d x$

Question Number 67197 Answers: 1 Comments: 0

∫_0 ^2 x^5 (1−(x/2))^4 dx

$\int_{0}^{2} x^{5} {(1 - \frac{x}{2})}^{4} d x$

Question Number 67153 Answers: 2 Comments: 0

find ∫(v^3 −2)/(v^4 +v )dv

$f i n d \int (v^{3} - 2) / (v^{4} + v) d v$

Question Number 67138 Answers: 0 Comments: 1

find the area abounded y=(√x) and y=x−2?

$f i n d t h e a r e a a b o u n d e d y = \sqrt{x}$ $a n d y = x - 2 ?$

Question Number 67106 Answers: 0 Comments: 1

find the area abounded y=(√x) afind y=x−2?

$f i n d t h e a r e a a b o u n d e d y = \sqrt{x}$ $a f i n d y = x - 2 ?$

Question Number 67105 Answers: 0 Comments: 0

find the area abounded y=(√x) afind y=x−2?

$f i n d t h e a r e a a b o u n d e d y = \sqrt{x}$ $a f i n d y = x - 2 ?$

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