Let -dx-x-2008-x-1-p-ln - x-q-1-x-r -C where p-q-r - N and need not be distinct- then the value of -p-q-r- equals 6024-6022-6021-6020-

The correct option is C 6021 I = ∫dx/x(x^2007+1) = ∫x^2007+1 x^2007/x(x^2007+1)dx = ∫ ( 1/x x^2006/1+x^2007 )dx =ln x 1/2007 ln(1+x^2007) =ln x^2007 ln(1+x^2 ...

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Byju's Answer

Standard XII

Mathematics

Integration by Partial Fractions

Let ∫dx/x^200...

Question

Let $\int \frac{d x}{x^{2008} + x} = \frac{1}{p} l n (\frac{x^{q}}{1 + x^{r}}) + C$ where $p, q, r ϵ N$ and need not be distinct, then the value of (p+q+r) equals

6024

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6022

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6021

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6020

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Solution

The correct option is C
6021

$I = \int \frac{d x}{x (x^{2007} + 1)} = \int \frac{x^{2007} + 1 - x^{2007}}{x (x^{2007} + 1)} d x$
$= \int (\frac{1}{x} - \frac{x^{2006}}{1 + x^{2007}}) d x$
$= l n x - \frac{1}{2007} l n (1 + x^{2007})$
$= \frac{l n x^{2007} - l n (1 + x^{2007})}{2007}$
$= \frac{1}{2007} l n (\frac{x^{2007}}{1 + x^{2007}}) + C$
$∴ p + q + r = 6021$

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Let ∫dxx2008+x=1p ln(xq1+xr)+C where p,q,rϵN and need not be distinct, then the value of (p+q+r) equals

The correct option is C 6021 I=∫dxx(x2007+1)=∫x2007+1−x2007x(x2007+1)dx =∫(1x−x20061+x2007)dx =ln x−12007 ln(1+x2007) =ln x2007−ln(1+x2007)2007 =12007 ln(x20071+x2007)+C ∴p+q+r=6021

Let $\int \frac{d x}{x^{2008} + x} = \frac{1}{p} l n (\frac{x^{q}}{1 + x^{r}}) + C$ where $p, q, r ϵ N$ and need not be distinct, then the value of (p+q+r) equals