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IntegrationQuestion and Answers: Page 301

Question Number 31839 Answers: 0 Comments: 1

I = ∫ (√(x + (√(x^2 − 1)))) dx

$I = \int \sqrt{x + \sqrt{x^{2} - 1}} d x$

Question Number 31838 Answers: 0 Comments: 0

Given f(x) = (3/(16)) (∫_0 ^1 f(x)dx)x^2 − (9/(10))(∫_0 ^2 f(x)dx)x + 2(∫_0 ^3 f(x)dx) + 4 Solve lim_(t→0) ((2t + (∫_(f(2) + 2) ^(f^(−1) (t)) [f ′(x)]^2 dx))/(1 − cos t cosh 2t cos 3t))

$Given$ $f (x) = \frac{3}{16} (\int_{0}^{1} f (x) d x) x^{2} - \frac{9}{10} (\int_{0}^{2} f (x) d x) x + 2 (\int_{0}^{3} f (x) d x) + 4$ $Solve$ $\lim_{t \to 0} \frac{2 t + (\int_{f (2) + 2}^{f^{- 1} (t)} {[f^{'} (x)]}^{2} d x)}{1 - \cos t \cosh 2 t \cos 3 t}$

Question Number 31787 Answers: 2 Comments: 0

∫((4x−3)/(x^2 +3x+8))dx

$\int \frac{4 x - 3}{x^{2} + 3 x + 8} d x$

Question Number 31747 Answers: 0 Comments: 1

let give ∣λ∣<1 and u_n = ∫_0 ^π ((cos(nx))/(1−2λ cosx +λ^2 )) prove that Σ_(n=0) ^∞ u_n is convergent and find its sum .

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

Question Number 31517 Answers: 0 Comments: 1

find ∫_(−1) ^1 (dx/((√(1+x)) +(√(1−x)))) .

$f i n d \int_{- 1}^{1} \frac{d x}{\sqrt{1 + x} + \sqrt{1 - x}} .$

Question Number 31516 Answers: 1 Comments: 1

find ∫ (dx/(x +(√(1+x^2 )))) .

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

Question Number 31515 Answers: 1 Comments: 1

calculate ∫_0 ^1 (dx/(chx)) .

$c a l c u l a t e \int_{0}^{1} \frac{d x}{c h x} .$

Question Number 31514 Answers: 1 Comments: 0

find ∫_0 ^1 ((arctan(2x))/((1+x)^2 ))dx.

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

Question Number 31513 Answers: 1 Comments: 1

find ∫_0 ^(2π) (dx/(2 +cosx)) .

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

Question Number 31512 Answers: 0 Comments: 1

find lim_(x→∞) ∫_x ^(2x) ((cos((1/t)))/t) dt.

$f i n d {l i m}_{x \to \infty} \int_{x}^{2 x} \frac{c o s (\frac{1}{t})}{t} d t .$

Question Number 31507 Answers: 0 Comments: 0

g is real function continue let f(x)=∫_0 ^x sin(x−t)g(t)dt 1)prove that f^′ (x)= ∫_0 ^x cos(t−x)g(t)dt 2)prove that f is so<ution of the diff.equa. y^(′′) +y =g(x)

$g i s r e a l f u n c t i o n c o n t i n u e l e t$ $f (x) = \int_{0}^{x} s i n (x - t) g (t) d t$ $1) p r o v e t h a t f^{^{'}} (x) = \int_{0}^{x} c o s (t - x) g (t) d t$ $2) p r o v e t h a t f i s s o < u t i o n o f t h e d i f f . e q u a .$ $y^{^{″}} + y = g (x)$

Question Number 31506 Answers: 0 Comments: 1

let f(x)=∫_x ^(2x) ((sht)/t)dt 1) calculate f^′ (x) 2) find lim_(x→0) f(x) .

$l e t f (x) = \int_{x}^{2 x} \frac{s h t}{t} d t$ $1) c a l c u l a t e f^{^{'}} (x)$ $2) f i n d {l i m}_{x \to 0} f (x) .$

Question Number 31505 Answers: 0 Comments: 0

find ∫_a ^b ((1−x^2 )/((1+x^2 )(√(1+x^4 ))))dx with a>1 and b>1.

$f i n d \int_{a}^{b} \frac{1 - x^{2}}{(1 + x^{2}) \sqrt{1 + x^{4}}} d x w i t h a > 1 a n d b > 1 .$

Question Number 31504 Answers: 0 Comments: 1

calculate ∫_0 ^1 (dt/(t +(√(1−t^2 )))) .

$c a l c u l a t e \int_{0}^{1} \frac{d t}{t + \sqrt{1 - t^{2}}} .$

Question Number 31503 Answers: 0 Comments: 1

find ∫_2 ^(√5) (dt/(t(√(t^2 −1)))) .

$f i n d \int_{2}^{\sqrt{5}} \frac{d t}{t \sqrt{t^{2} - 1}} .$

Question Number 31501 Answers: 0 Comments: 1

find ∫_0 ^(π/4) ln(1 +2tanx)dx.

$f i n d \int_{0}^{\frac{π}{4}} l n (1 + 2 t a n x) d x .$

Question Number 31466 Answers: 0 Comments: 0

let give I_n = ∫_(1/n) ^1 (√(1+t^2 )) dt 1) calculate I_n 2) find lim_(n→∞) I_n .

$l e t g i v e I_{n} = \int_{\frac{1}{n}}^{1} \sqrt{1 + t^{2}} d t$ $1) c a l c u l a t e I_{n}$ $2) f i n d {l i m}_{n \to \infty} I_{n} .$

Question Number 31465 Answers: 0 Comments: 0

find F(α)= ∫_0 ^1 ((arctan(αx))/(1+x^2 )) dx with α ∈ R−{1,−1}

$f i n d F (α) = \int_{0}^{1} \frac{a r c t a n (α x)}{1 + x^{2}} d x w i t h α \in R - {1, - 1}$

Question Number 31464 Answers: 0 Comments: 0

1) find A_n = ∫_0 ^(π/2) e^(−x) cos(nx)dx 2) find S_n = Σ_(k=0) ^n A_k .

$1) f i n d A_{n} = \int_{0}^{\frac{π}{2}} e^{- x} c o s (n x) d x$ $2) f i n d S_{n} = \sum_{k = 0}^{n} A_{k} .$

Question Number 31463 Answers: 0 Comments: 0

let give the function f(x)=∫_0 ^π ln(1+xcosθ)dθ with ∣x∣<1 1) find a simple form of f(x) 2)calculate ∫_0 ^π ln(1−cosθ)dθ 3)calculate ∫_0 ^π ln(1+cosθ)dθ.

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