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IntegrationQuestion and Answers: Page 243

Question Number 54372 Answers: 1 Comments: 3

let f(x) =∫_0 ^(2π) ((sint)/(x+sint))dt 1) calculate f(x) 2) calculate g(x) =∫_0 ^(2π) ((sint)/((x+sint)^2 )) dt 3) calculste for n∈N ∫_0 ^(2π) ((sint)/((x+sint)^n ))dt 4) calculate ∫_0 ^(2π) ((sint)/(2+sint))dt and ∫_0 ^(2π) ((sint)/((2+sint)^2 ))dt .

$l e t f (x) = \int_{0}^{2 π} \frac{s i n t}{x + s i n t} d t$ $1) c a l c u l a t e f (x)$ $2) c a l c u l a t e g (x) = \int_{0}^{2 π} \frac{s i n t}{{(x + s i n t)}^{2}} d t$ $3) c a l c u l s t e f o r n \in N \int_{0}^{2 π} \frac{s i n t}{{(x + s i n t)}^{n}} d t$ $4) c a l c u l a t e \int_{0}^{2 π} \frac{s i n t}{2 + s i n t} d t a n d \int_{0}^{2 π} \frac{s i n t}{{(2 + s i n t)}^{2}} d t .$

Question Number 54371 Answers: 1 Comments: 1

prove that ln(z) = ∫_0 ^1 ((z−1)/(1+t(z−1)))dt .

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

Question Number 54367 Answers: 1 Comments: 1

find ∫_1 ^(+∞) (([t])/t) t^(−p) dt interms of ξ(p) with p>0 .

$f i n d \int_{1}^{+ \infty} \frac{[t]}{t} t^{- p} d t i n t e r m s o f ξ (p) w i t h p > 0 .$

Question Number 54259 Answers: 1 Comments: 1

Question Number 54273 Answers: 2 Comments: 0

Question Number 54248 Answers: 2 Comments: 1

Question Number 54240 Answers: 1 Comments: 1

Question Number 54224 Answers: 2 Comments: 4

∫_0 ^( (π/4)) ((sinx+cosx)/(16+9sin2x)) dx =?

$\int_{0}^{\frac{π}{4}} \frac{\sin x + \cos x}{16 + 9 \sin 2 x} d x = ?$

Question Number 54209 Answers: 2 Comments: 0

Question Number 54102 Answers: 1 Comments: 3

the absolute value ∫_(10) ^(19) ((cos x)/(1+x^8 )) dx is...

$the absolute value$ $\int_{10}^{19} \frac{\cos x}{1 + x^{8}} d x is . . .$

Question Number 54074 Answers: 1 Comments: 5

Evaluate : 1) ∫_0 ^( 1) (dx/((√(1+x))+(√(1−x))+2)) 2) ∫_0 ^( 2) ((ln(1+2x))/(1+x^2 )) 3) ∫_0 ^( π) (x/(√(1+sin^3 x)))((3πcosx+4sinx)sin^2 x+4)dx 4) ∫_0 ^( π) ((x^2 cos^2 x−xsinx−cosx−1)/((1+xsinx)^2 )) dx.

$E v a l u a t e :$ $1) \int_{0}^{1} \frac{d x}{\sqrt{1 + x} + \sqrt{1 - x} + 2}$ $2) \int_{0}^{2} \frac{l n (1 + 2 x)}{1 + x^{2}}$ $3)$ $\int_{0}^{π} \frac{x}{\sqrt{1 + \sin^{3} x}} ((3 π \cos x + 4 \sin x) \sin^{2} x + 4) d x$ $4)$ $\int_{0}^{π} \frac{x^{2} \cos^{2} x - x \sin x - \cos x - 1}{{(1 + x \sin x)}^{2}} d x .$

Question Number 54073 Answers: 1 Comments: 0

Question Number 54070 Answers: 2 Comments: 7

Evaluate : 1) ∫_(−1) ^( 1) cot^(−1) ((1/(√(1−x^2 )))).(cot^(−1) (x/(√(1−(x^2 )^(∣x∣) ))))dx 2) ∫_0 ^( (π/2)) ((sin^2 (10)θ)/(sin^2 θ)) dθ 3) ∫_0 ^( (π/4)) ((ln(cotx))/(((sinx)^(2009) +(cosx)^(2009) )^2 )).(sin2x)^(2008) dx 4) ∫_0 ^( 2) ((4x+10)/((x^2 +5x+6)^2 )) dx.

$E v a l u a t e :$ $1)$ $\int_{- 1}^{1} \cot^{- 1} (\frac{1}{\sqrt{1 - x^{2}}}) . (co t^{- 1} \frac{x}{\sqrt{1 - {(x^{2})}^{∣ x ∣}}}) d x$ $2)$ $\int_{0}^{\frac{π}{2}} \frac{\sin^{2} (10) θ}{\sin^{2} θ} d θ$ $3)$ $\int_{0}^{\frac{π}{4}} \frac{l n (c o t x)}{{({(\sin x)}^{2009} + {(\cos x)}^{2009})}^{2}} . {(\sin 2 x)}^{2008} d x$ $4)$ $\int_{0}^{2} \frac{4 x + 10}{{(x^{2} + 5 x + 6)}^{2}} d x .$

Question Number 54011 Answers: 1 Comments: 2

calculate f(a) =∫ (dx/((√(1+ax))−(√(1−ax)))) with a>0 . 2) calculate U_n =∫_(−(1/(na))) ^(1/(na)) (dx/((√(1+ax))−(√(1−ax)))) with n from N and n>1 find lim_(n→+∞) U_n and study the convergence of Σ U_n

$c a l c u l a t e f (a) = \int \frac{d x}{\sqrt{1 + a x} - \sqrt{1 - a x}} w i t h a > 0 .$ $2) c a l c u l a t e U_{n} = \int_{- \frac{1}{n a}}^{\frac{1}{n a}} \frac{d x}{\sqrt{1 + a x} - \sqrt{1 - a x}} w i t h n f r o m N a n d n > 1$ $f i n d {l i m}_{n \to + \infty} U_{n} a n d s t u d y t h e c o n v e r g e n c e o f Σ U_{n}$