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IntegrationQuestion and Answers: Page 233

Question Number 61328 Answers: 0 Comments: 4

let f(a) =∫_0 ^1 ((sin(2x))/(1+ax^2 )) dx with ∣a∣<1 1) approximate f(a) by a polynom 2) find the value (perhaps not exact) of ∫_0 ^1 ((sin(2x))/(1+2x^2 )) dx 3) let g(a) = ∫_0 ^1 ((x^2 sin(2x))/((1+ax^2 )^2 )) dx approximat g(a) by a polynom 4) find the value of ∫_0 ^1 ((x^2 sin(2x))/((1+2x^2 )^2 )) dx .

$l e t f (a) = \int_{0}^{1} \frac{s i n (2 x)}{1 + {a x}^{2}} d x w i t h ∣ a ∣< 1$ $1) a p p r o x i m a t e f (a) b y a p o l y n o m$ $2) f i n d t h e v a l u e (p e r h a p s n o t e x a c t) o f \int_{0}^{1} \frac{s i n (2 x)}{1 + 2 x^{2}} d x$ $3) l e t g (a) = \int_{0}^{1} \frac{x^{2} s i n (2 x)}{{(1 + {a x}^{2})}^{2}} d x a p p r o x i m a t g (a) b y a p o l y n o m$ $4) f i n d t h e v a l u e o f \int_{0}^{1} \frac{x^{2} s i n (2 x)}{{(1 + 2 x^{2})}^{2}} d x .$

Question Number 61326 Answers: 0 Comments: 4

find ∫_0 ^1 ((sinx)/(1+x^2 ))dx

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

Question Number 61240 Answers: 1 Comments: 3

∫ ((x^(2 ) − 4)/((x^2 + 4)^2 )) dx

$\int \frac{x^{2} - 4}{{(x^{2} + 4)}^{2}} dx$

Question Number 61237 Answers: 0 Comments: 0

Question Number 61232 Answers: 0 Comments: 3

let U_n =∫_1 ^(+∞) (([nx]−[(n−1)x])/x^3 ) dx with n≥1 1) find U_n interms of n 2) find lim_(n→+∞) U_n 3) study the serie Σ_(n=1) ^∞ U_n

$l e t U_{n} = \int_{1}^{+ \infty} \frac{[n x] - [(n - 1) x]}{x^{3}} d x w i t h n ⩾ 1$ $1) f i n d U_{n} i n t e r m s o f n$ $2) f i n d {l i m}_{n \to + \infty} U_{n}$ $3) s t u d y t h e s e r i e \sum_{n = 1}^{\infty} U_{n}$

Question Number 61229 Answers: 1 Comments: 0

let f_n (a) =∫_0 ^a x^n (√(a^2 −x^2 ))dx with a>0 1) determine a explicit form of f(a) 2) let g_n (a) =f^′ (a) give g_n (a) at form of integral and give its value 3) find the value of ∫_0 ^2 x^3 (√(4−x^2 ))dx and ∫_0 ^(√3) x^4 (√(3−x^2 ))dx

$l e t f_{n} (a) = \int_{0}^{a} x^{n} \sqrt{a^{2} - x^{2}} d x w i t h a > 0$ $1) d e t e r m i n e a e x p l i c i t f o r m o f f (a)$ $2) l e t g_{n} (a) = f^{^{'}} (a) g i v e g_{n} (a) a t f o r m o f i n t e g r a l a n d g i v e i t s$ $v a l u e$ $3) f i n d t h e v a l u e o f \int_{0}^{2} x^{3} \sqrt{4 - x^{2}} d x a n d \int_{0}^{\sqrt{3}} x^{4} \sqrt{3 - x^{2}} d x$

Question Number 61208 Answers: 0 Comments: 8

Question Number 61056 Answers: 2 Comments: 0

∫ ((x + sin(x))/(1 + cos(x))) dx

$\int \frac{x + \sin (x)}{1 + \cos (x)} dx$

Question Number 61045 Answers: 2 Comments: 2

calculate I =∫_0 ^1 cos(2arctanx)dx and J =∫_0 ^1 sin(2arctanx)dx

$c a l c u l a t e I = \int_{0}^{1} c o s (2 a r c t a n x) d x$ $a n d J = \int_{0}^{1} s i n (2 a r c t a n x) d x$

Question Number 61041 Answers: 1 Comments: 1

calculate ∫_1 ^(+∞) (([2x]−[x])/x^4 ) dx

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

Question Number 61039 Answers: 0 Comments: 1

find ∫_0 ^1 arctan((2/(1+x)))dx

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

Question Number 60976 Answers: 0 Comments: 1

find Σ_(n=1) ^∞ (1/n^2 ) by use of integral ∫_0 ^(π/2) ln(2cosθ)dθ .

$f i n d \sum_{n = 1}^{\infty} \frac{1}{n^{2}} b y u s e o f i n t e g r a l \int_{0}^{\frac{π}{2}} l n (2 c o s θ) d θ .$

Question Number 60967 Answers: 0 Comments: 4

study the integral ∫_(−∞) ^(+∞) (1−cos((2/(x^2 +1))))dx

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

Question Number 60960 Answers: 0 Comments: 2

find ∫_(−∞) ^(+∞) tan((1/(1+x^2 )))dx

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

Question Number 60944 Answers: 1 Comments: 1

∫(dx/((1+x^2 )^(3/2) )) solve this pls

$\int \frac{d x}{{(1 + x^{2})}^{\frac{3}{2}}}$ $s o l v e t h i s p l s$

Question Number 60938 Answers: 0 Comments: 1

∫((csc^(2019) (x))/(sec^5 (x))) tan^2 (x) dx

$\int \frac{{c s c}^{2019} (x)}{{s e c}^{5} (x)} {t a n}^{2} (x) d x$

Question Number 60901 Answers: 1 Comments: 0

find ∫ arctan((1/(1+x^2 )))dx

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

Question Number 60894 Answers: 0 Comments: 0

study the convergence of ∫_0 ^1 (((√(1+2x))−(√(1+x)))/(ln(1+x)))dx and determine its value.

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

Question Number 60893 Answers: 1 Comments: 1

calculate ∫_0 ^(π/2) (dx/((√2)cos^2 x +(√3)sin^2 x))

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

Question Number 60881 Answers: 0 Comments: 3

∫_(−π) ^π sin((1/(1−x^2 ))) dx

$\underset{- π}{\int^{π}} s i n (\frac{1}{1 - x^{2}}) d x$

Question Number 60797 Answers: 0 Comments: 2

∫(e^w /w^(n+1) )dw, n∈N

$\int \frac{e^{w}}{w^{n + 1}} d w, n \in N$

Question Number 60791 Answers: 1 Comments: 2

∫(e^n /x^(n+1) )dx, n∈N

$\int \frac{e^{n}}{x^{n + 1}} d x, n \in N$

Question Number 60783 Answers: 0 Comments: 2

∫_(−∞) ^∞ sin((1/(1+x^2 ))) dx

$\underset{- \infty}{\int^{\infty}} s i n (\frac{1}{1 + x^{2}}) d x$

Question Number 60739 Answers: 1 Comments: 4

evaluate i.∫ (((x+1)/(x−1)))dx ii. ∫_0 ^π (2cosxsinx)dx iii. ∫_((π/(3 )) ) ^π (((sin2x)/(cos2x)))dx

$e v a l u a t e$ $i . \int (\frac{x + 1}{x - 1}) d x$ $i i . \int_{0}^{π} (2 c o s x s i n x) d x$ $i i i . \int_{\frac{π}{3}}^{π} (\frac{s i n 2 x}{c o s 2 x}) d x$

Question Number 60731 Answers: 0 Comments: 0

Question Number 60728 Answers: 0 Comments: 0

calculate ∫_1 ^(+∞) ((ln(lnx))/(x^2 −x +1))dx

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

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