If 2 x 12 + 5 x 9 ( 1 + x 3 + x 5 ) 3 d x = x m ( 1 + x 3 + x 5 ) r + C , then m + r =

Question

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 $\frac{m + ℓ}{r} =$

Smart Achievers · Accepted Answer

(4)

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

Put $\frac{1}{x^{2}} + \frac{1}{x^{5}} + 1 = t \Rightarrow \int \frac{- d t}{t^{3}} = \frac{1}{2 t^{2}} + c = \frac{1}{2 {(\frac{1}{x^{5}} + \frac{1}{x^{2}} + 1)}^{2}} + c = \frac{x^{10}}{2 {(1 + x^{3} + x^{5})}^{2}} + c$

$m = 10; r = 2; l = 2; \frac{m + l}{r} = 6$

Solution

Q.
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 $\frac{m + ℓ}{r} =$

1 3

2 4

3 5

4 6

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Solution

Q. If ∫2x12+5x9(1+x3+x5)3dx=xmℓ(1+x3+x5)r+C, then m+ℓr=

1 3

2 4

3 5

4 6

Ans. (4) ∫2x12+5x9(1+x+x5)3dx=∫2x12+5x9x15(1x5+1x2+1)3dx=∫2x3+5x6(1x5+1x2+1)3dx Put 1x2+1x5+1=t⇒∫-dtt3=12t2+c=12(1x5+1x2+1)2+c=x102(1+x3+x5)2+c m=10; r=2; l=2; m+lr=6

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Q.
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 $\frac{m + ℓ}{r} =$