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Question Number 38130 Answers: 4 Comments: 0

1. ∫tan^3 (2x)sec^5 (2x) dx 2. ∫_0 ^(π/3) tan^5 (x)sec^6 (x) dx 3. ∫tan^6 (ay) dy

$1 . \int \tan^{3} (2 x) \sec^{5} (2 x) d x$ $2 . \int_{0}^{\frac{π}{3}} \tan^{5} (x) \sec^{6} (x) d x$ $3 . \int \tan^{6} (a y) d y$

Question Number 38127 Answers: 0 Comments: 1

calculate ∫_0 ^∞ e^(−x) (√(1+e^(−2x) ))dx

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

Question Number 38126 Answers: 1 Comments: 2

calculate ∫_0 ^∞ e^(−2x) (√(1+e^(−4x) ))dx .

$c a l c u l a t e \int_{0}^{\infty} e^{- 2 x} \sqrt{1 + e^{- 4 x}} d x .$

Question Number 38125 Answers: 1 Comments: 1

let α>0 find ∫_0 ^∞ (e^(−αx) /(√(1+e^(−2αx) )))dx .

$l e t α > 0 f i n d$ $\int_{0}^{\infty} \frac{e^{- α x}}{\sqrt{1 + e^{- 2 α x}}} d x .$

Question Number 38124 Answers: 1 Comments: 0

prove that ∫ (dx/(√(1+x^2 ))) =ln(x+(√(1+x^2 ))) +c 2) find ∫ (dx/(√(a+x^2 ))) with a>0

$p r o v e t h a t \int \frac{d x}{\sqrt{1 + x^{2}}} = l n (x + \sqrt{1 + x^{2}}) + c$ $2) f i n d \int \frac{d x}{\sqrt{a + x^{2}}} w i t h a > 0$

Question Number 38123 Answers: 2 Comments: 1

f is a function positive and C^1 1) find ∫ (f^′ /(2(√f)(√(1+f))))dx 2)let A_n = ∫_0 ^1 (x^(n/2) /(x(√(1+x^n )))) calculate A_n and lim_(n→+∞) A_n

$f i s a f u n c t i o n p o s i t i v e a n d C^{1}$ $1) f i n d \int \frac{f^{^{'}}}{2 \sqrt{f} \sqrt{1 + f}} d x$ $2) l e t A_{n} = \int_{0}^{1} \frac{x^{\frac{n}{2}}}{x \sqrt{1 + x^{n}}}$ $c a l c u l a t e A_{n} a n d {l i m}_{n \to + \infty} A_{n}$

Question Number 38122 Answers: 1 Comments: 0

calculate ∫_0 ^π ((sinx)/(√(1+cos^2 x)))dx

$c a l c u l a t e \int_{0}^{π} \frac{s i n x}{\sqrt{1 + {c o s}^{2} x}} d x$

Question Number 38121 Answers: 0 Comments: 5

let x>0 find F(x) = ∫_(−∞) ^(+∞) ((arctan(xt^2 ))/(1+t^2 ))dt

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

Question Number 38120 Answers: 0 Comments: 2

let n from N and find the value of A_n = ∫_1 ^(+∞) (dt/(t^n (√(t−1))))

$l e t n f r o m N a n d$ $f i n d t h e v a l u e o f A_{n} = \int_{1}^{+ \infty} \frac{d t}{t^{n} \sqrt{t - 1}}$

Question Number 38119 Answers: 0 Comments: 1

calculate ∫_1 ^(+∞) (dx/(x^4 (√(x−1))))

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

Question Number 38118 Answers: 0 Comments: 1

prove that ∫_0 ^1 (1/(1+(t^a /2)))dt =Σ_(n=0) ^∞ (((−1)^n )/(2^n (na+1))) 2) find the value of Σ_(n=0) ^∞ (((−1)^n )/(2^n (3n+1)))

$p r o v e t h a t \int_{0}^{1} \frac{1}{1 + \frac{t^{a}}{2}} d t = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{2^{n} (n a + 1)}$ $2) f i n d t h e v a l u e o f \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{2^{n} (3 n + 1)}$

Question Number 38117 Answers: 0 Comments: 0

let x and y from R prove that ∣cos(x+iy)∣=cos^2 x +sh^2 y ∣sin(x+iy)∣^2 =sin^2 x +sh^2 y

$l e t x a n d y f r o m R p r o v e t h a t$ $∣ c o s (x + i y) ∣= {c o s}^{2} x + {s h}^{2} y$ $∣ s i n (x + i y) ∣^{2} = {s i n}^{2} x + {s h}^{2} y$

Question Number 38116 Answers: 0 Comments: 1

find I = ∫_0 ^∞ ((cos(λx))/(ch(2x)))dx

$f i n d I = \int_{0}^{\infty} \frac{c o s (λ x)}{c h (2 x)} d x$

Question Number 38115 Answers: 0 Comments: 0

find ∫_0 ^∞ ((sin(2x))/(sh(3x)))dx

$f i n d \int_{0}^{\infty} \frac{s i n (2 x)}{s h (3 x)} d x$

Question Number 38114 Answers: 0 Comments: 2

let I_n = ∫_0 ^(2π) (dx/((p +cost)^n )) with p>1 find the value of I_n

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

Question Number 38113 Answers: 0 Comments: 2

let p>1 calculate ∫_0 ^(2π) (dt/((p +cost)^2 ))

$l e t p > 1 c a l c u l a t e \int_{0}^{2 π} \frac{d t}{{(p + c o s t)}^{2}}$

Question Number 38112 Answers: 1 Comments: 1

prove that arctan(x)= (i/2)ln(((i+x)/(i−x))) for ∣x∣<1

$p r o v e t h a t a r c t a n (x) = \frac{i}{2} l n (\frac{i + x}{i - x}) f o r ∣ x ∣< 1$

Question Number 38111 Answers: 1 Comments: 1

find lim_(x→0) ((e^x −[x])/x)

$f i n d {l i m}_{x \to 0} \frac{e^{x} - [x]}{x}$

Question Number 38110 Answers: 0 Comments: 0

let x from R find the value of f(x)= ∫_0 ^π ln(x^2 −2x cosθ +1)dθ

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

Question Number 38109 Answers: 0 Comments: 2

1) find S(x) = Σ_(n=1) ^∞ ((cos(nx))/n) 2) find Σ_(n=1) ^∞ (((−1)^n )/n)

$1) f i n d S (x) = \sum_{n = 1}^{\infty} \frac{c o s (n x)}{n}$ $2) f i n d \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{n}$

Question Number 38108 Answers: 0 Comments: 1

find Σ_(n=1) ^∞ (((−1)^n )/n^2 )

$f i n d \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{n^{2}}$

Question Number 38107 Answers: 0 Comments: 0

find C = Σ_(n=1) ^∞ ((cos(nx))/n^2 )dx and S=Σ_(n=1) ^∞ ((sin(nx))/n^2 )

$f i n d C = \sum_{n = 1}^{\infty} \frac{c o s (n x)}{n^{2}} d x a n d S = \sum_{n = 1}^{\infty} \frac{s i n (n x)}{n^{2}}$

Question Number 38106 Answers: 0 Comments: 1

calculate ∫_0 ^(+∞) e^(−3t) ln(1+e^t )dt .

$c a l c u l a t e \int_{0}^{+ \infty} e^{- 3 t} l n (1 + e^{t}) d t .$

Question Number 38105 Answers: 1 Comments: 0

find ∫ (dx/((√(2x+1)) +(√(2x−1))))

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

Question Number 38104 Answers: 1 Comments: 0

find ∫_1 ^(+∞) (dx/((x^2 +2)(√(x+3))))

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

Question Number 38103 Answers: 0 Comments: 0

find I(λ)= ∫_0 ^(π/2) ((xdx)/(λ +tanx)) λ from R.

$f i n d I (λ) = \int_{0}^{\frac{π}{2}} \frac{x d x}{λ + t a n x} λ f r o m R .$

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