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IntegrationQuestion and Answers: Page 10

Question Number 207949 Answers: 2 Comments: 0

Find the value of : 𝛀 = ∫_0 ^( (𝛑/2)) (( dx)/(sin^6 x + cos^6 x)) = ? −−−−−−−−−

$F i n d t h e v a l u e o f :$ $Ω = \int_{0}^{\frac{π}{2}} \frac{d x}{{s i n}^{6} x + {c o s}^{6} x} = ?$ $- - - - - - - - -$

Question Number 207938 Answers: 1 Comments: 0

what is the area bounded by the curve y=x(x−2)(x−5) and the x axis?

$w h a t i s t h e a r e a b o u n d e d b y t h e c u r v e$ $y = x (x - 2) (x - 5) a n d t h e x a x i s ?$

Question Number 207924 Answers: 0 Comments: 0

∫f(x)g(x)dx=Σ_(n=0) ^∞ (−1)^n lim_(h→0) (1/h^n ) Σ_(i=o) ^n [ (−1)^i (((n!)/(i!(n−i)!)))f(x+(n−i)h)] (1/(n!))∫_a ^x (x−t)^n g(t)dt prove that right its a relation that i have derrived

$\int f (x) g (x) d x = \underset{n = 0}{\sum^{\infty}} {(- 1)}^{n} \lim_{h \to 0} \frac{1}{h^{n}} \underset{i = o}{\sum^{n}} [{(- 1)}^{i} (\frac{n!}{i! (n - i)!}) f (x + (n - i) h)] \frac{1}{n!} \underset{a}{\int^{x}} {(x - t)}^{n} g (t) d t$ $p r o v e t h a t r i g h t$ $i t s a r e l a t i o n t h a t i h a v e d e r r i v e d$

Question Number 207906 Answers: 1 Comments: 0

help ∫_1 ^( ∞) x^(−ln(x)) dx

$h e l p$ $\int_{1}^{\infty} x^{- l n (x)} d x$

Question Number 207878 Answers: 2 Comments: 0

Question Number 207857 Answers: 1 Comments: 0

∫xtan^(−1) xdx

$\int x \tan^{- 1} x d x$

Question Number 207789 Answers: 1 Comments: 0

∀r∈R: H_r =∫_0 ^1 ((t^r −1)/(t−1))dt H_(r+2) −H_r =1 r=?

$\forall r \in R : H_{r} = \underset{0}{\int^{1}} \frac{t^{r} - 1}{t - 1} d t$ $H_{r + 2} - H_{r} = 1$ $r = ?$

Question Number 207753 Answers: 1 Comments: 0

Question Number 207707 Answers: 0 Comments: 0

∫_0 ^(+∞) ((sin^2 (x))/(sin^2 (x)+(xcos (x)+sin (x))^2 ))d(x)

$\underset{0}{\int^{+ \infty}} \frac{\sin^{2} (x)}{\sin^{2} (x) + {(x \cos (x) + \sin (x))}^{2}} d (x)$

Question Number 207652 Answers: 1 Comments: 0

∫_0 ^1 log(1+x^3 )dx = ?and ∫_0 ^1 log (1+x^4 )dx = ? and if possible then find the value of p p = ∫_0 ^1 log(1+x^n )dx = ? n∈N

$\int_{0}^{1} \log (1 + x^{3}) d x = ? a n d \int_{0}^{1} \log (1 + x^{4}) d x = ?$ $and if possible then find the value of p$ $p = \int_{0}^{1} \log (1 + x^{n}) d x = ? n \in N$

Question Number 207582 Answers: 2 Comments: 0

∫_0 ^π ln(sinx)dx=−πln2 ∫_0 ^1 lnΓ(x)dx = ln(2π)

$\int_{0}^{π} l n (s i n x) d x = - π l n 2$ $\int_{0}^{1} l n Γ (x) d x = l n (2 π)$

Question Number 207565 Answers: 2 Comments: 0

find ∫_0 ^(π/2) (x^2 /(tan^2 x))dx

$f i n d \int_{0}^{\frac{π}{2}} \frac{x^{2}}{{t a n}^{2} x} d x$

Question Number 207424 Answers: 0 Comments: 0

f_n (x):=∫e^((2x)/3) ((cos(x))/( (cos(x)+sin(x))^(n/3) ))dx=...? for n=1, i found f_1 (x)=(3/4)e^((2x)/3) (cos(x)+sin(x))^(2/3) + C is there any ideas for a general case or the case n=2?

$f_{n} (x) := \int e^{\frac{2 x}{3}} \frac{c o s (x)}{{(c o s (x) + s i n (x))}^{\frac{n}{3}}} d x = . . . ?$ $f o r n = 1, i f o u n d$ $f_{1} (x) = \frac{3}{4} e^{\frac{2 x}{3}} {(c o s (x) + s i n (x))}^{\frac{2}{3}} + C$ $i s t h e r e a n y i d e a s f o r a g e n e r a l c a s e o r$ $t h e c a s e n = 2 ?$