If ∫2x12+5x91+x3+x53 dx=xml1+x3+x5r+c, then m−lr=___

∫2x12+5x9(1+x3+x5)3dx=∫2x12+5x9x151x5+1x2+13dx=∫2x3+5x61x5+1x2+13dxLet 1x5+1x2+1=t−5x6−3x3dx=dt=∫−dtt3=−∫1t3dt=−∫t−3dt=−t−2−2+c=12t2+c=121x5+1x2+12+c=x102(1+x3+x5)2+c∴xml(1+x3+x5)r+cm = 10; l =2; r =2m−lr=10−22=4

If ∫2x12+5x91+x3+x53 dx=xml1+x3+x5r+c, then m−lr=___

∫2x12+5x9(1+x3+x5)3dx=∫2x12+5x9x151x5+1x2+13dx=∫2x3+5x61x5+1x2+13dxLet 1x5+1x2+1=t−5x6−3x3dx=dt=∫−dtt3=−∫1t3dt=−∫t−3dt=−t−2−2+c=12t2+c=121x5+1x2+12+c=x102(1+x3+x5)2+c∴xml(1+x3+x5)r+cm = 10; l =2; r =2m−lr=10−22=4

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If $\int \frac{2 x^{12} + 5 x^{9}}{{(1 + x^{3} + x^{5})}^{3}} d x = \frac{x^{m}}{l {(1 + x^{3} + x^{5})}^{r}} + c$ , then $\frac{m - l}{r} =___$

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Detailed Solution

$\begin{array}{l} \int \frac{2 x^{12} + 5 x^{9}}{{(1 + x^{3} + x^{5})}^{3}} d x = \int \frac{2 x^{12} + 5 x^{9}}{x^{15} {(\frac{1}{x^{5}} + \frac{1}{x^{2}} + 1)}^{3}} d x \\ = \int \frac{\frac{2}{x^{3}} + \frac{5}{x^{6}}}{{(\frac{1}{x^{5}} + \frac{1}{x^{2}} + 1)}^{3}} d x \end{array}$

Let $\frac{1}{x^{5}} + \frac{1}{x^{2}} + 1 = t$

$\begin{array}{l} (\frac{- 5}{x^{6}} - \frac{3}{x^{3}}) d x = d t \\ = \int \frac{- d t}{t^{3}} \\ = - \int \frac{1}{t^{3}} d t \\ = - \int t^{- 3} d t \\ = - \frac{t^{- 2}}{- 2} + c \\ = \frac{1}{2 t^{2}} + c \end{array}$

$\begin{array}{l} = \frac{1}{2 {(\frac{1}{x^{5}} + \frac{1}{x^{2}} + 1)}^{2}} + c = \frac{x^{10}}{2 {(1 + x^{3} + x^{5})}^{2}} + c \\ ∴ \frac{x^{m}}{l {(1 + x^{3} + x^{5})}^{r}} + c \end{array}$