Calculus - very basic integral

JOHNCENA IS BACK ·Mar 20 '10 at 8:22

write in parametric form! denominator becomes sin(x+aplha)

take x+alpha=t and write x=t-alpha.......expand numerator n integrate!

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Pritish Chakraborty ·Mar 20 '10 at 10:54

(âˆšcot x) is damn easy job :P

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" ____________ ·Mar 20 '10 at 20:59

1. integrate $\int\frac{x^4 + 4 x ^3 +6x^2 + 4x + 1}{x^3 - 3x^2+ 3x -1} dx$

2. $\int ( sinx .log x - \frac{cosx}{x} ) dx$

FOR PRCTC

as alwayz pritish babu has done a nice work !!!!!!!!!!!!!!!!!

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Manmay kumar Mohanty ·Mar 20 '10 at 21:12
2) $\int (s i n x l o g x - x c o s x ) d x = \int s i n x l o g x d x - \int x c o s x d x$ .......(1)
let I₁ = $\int s i n x l o g x d x$
considering sin x as 2nd and log x as first function we have BY USING BY PARTS,
I₁ = $l o g x \int s i n x + \int x c o s x $
now substituting the value of I₁ in (1) we get,
$\int (s i n x l o g x - x c o s x ) d x =$ $l o g x \int s i n x + \int x c o s x $ - âˆ«cos xx = $l o g x \int s i n x$
= - logx cosx

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Manmay kumar Mohanty ·Mar 21 '10 at 0:51
LONG ONE THOUGH.
$\int (\sqrt t a n x + \sqrt c o t x ) d x = \int \sqrt s i n x c o s x s i n x + c o s x d x$ now put sinx - cosx = t â†’ (cosx + sinx)dx = dt
integral reduces to $\sqrt 2 \int \sqrt 1 - t 2 d t = \sqrt 2 s i n - 1 t + c = \sqrt 2 s i n - 1 (s i n x - c o s x) + c$
similarly we find $\int (\sqrt t a n x - \sqrt c o t x ) d x = - \sqrt 2 l n ∣ ∣ s i n x + c o s x + \sqrt s i n 2 x ∣ ∣ + c$
now $\int \sqrt c o t x = 2 1 \int 2 \sqrt c o t x d x = 2 1 \int (\sqrt t a n x + \sqrt c o t x ) - (\sqrt t a n x - \sqrt c o t x ) d x$
subtituting the values of $\int (\sqrt t a n x + \sqrt c o t x ) d x$ and $\int (\sqrt t a n x - \sqrt c o t x ) d x$ we get $\int \sqrt c o t x $ = $ 2 1 [\sqrt 2 s i n - 1 (s i n x - c o s x) + \sqrt 2 l n ∣ ∣ (s i n x + c o s x) + \sqrt s i n 2 x ∣ ∣ ] + k, k = i n t e g . c o n s t a n t$

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" ____________ ·Mar 21 '10 at 3:02
excellent work , manmay dada, & rahul

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sriraghav ·Mar 21 '10 at 8:49
whats the solution for 1st question ,
iam getting in the for of
K âˆ«√1- tan Î±. cos t dt
where t is x - Î±

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Unicorn--- Extinct!! ·Mar 22 '10 at 2:11
@ pritish...u pinked ur post urself?? [5][3]

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Manmay kumar Mohanty ·Mar 22 '10 at 3:05
uska paper hi pink hai yaar......

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arsh sharma ·Mar 22 '10 at 3:09
√cotx
yaar ise poora toh kar...

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Manmay kumar Mohanty ·Mar 22 '10 at 3:11
kiya to hai. PRITISH NE AUR MAINE

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