The value of ∫ ((x2-1) d x/x3 √2 x4-2 x2+1) d x is

Question

The value of ∫ ((x2-1) d x/x3 √2 x4-2 x2+1) d x is (A) 2 √2-(2/x2)+(1/x4)+c (B) 2 √2+(2/x2)+(1/x4)+c (C) (1/2) √2-(2/x2)+(1/x4)+c (D) none of these

Tardigrade Admin · Accepted Answer

The given integral is ∫ (x2-1/x3 √2 x4-2 x2+1) d x =∫ (x2-1/x5 √2-(2)x2+(1)x4 d x =∫ (((1/x3)-(1/x5)) d x/√2-(2)x2+(1)x4 =(1/2) ∫ (t d t/t), Let 2-(2/x2)+(1/x4)=t/2) ⇒ ((2/25)-(4/25)) d x=2 t d t =(1 / 2) t+c =(1/2) √/2-(2)x2+(1)x4+c

Q. The value of $\int \frac{( x ^{2} - 1 ) d x}{x ^{3} 2 x ^{4} - 2 x ^{2} + 1} d x$ is

$2 2 - \frac{2}{x ^{2}} + \frac{1}{x ^{4}} + c$

$2 2 + \frac{2}{x ^{2}} + \frac{1}{x ^{4}} + c$

$\frac{1}{2} 2 - \frac{2}{x ^{2}} + \frac{1}{x ^{4}} + c$

none of these

Q. The value of ∫x32x4−2x2+1​(x2−1)dx​dx is

22−x22​+x41​​+c

22+x22​+x41​​+c

21​2−x22​+x41​​+c

none of these

The given integral is ∫x32x4−2x2+1​x2−1​dx =∫x52−x22​+x41​​x2−1​dx =∫2−x22​+x41​​(x31​−x51​)dx​ =21​∫ttdt​, Let 2−x22​+x41​=t2 ⇒(252​−254​)dx=2tdt =(1/2)t+c =21​2−x22​+x41​​+c

Q. The value of $\int \frac{( x ^{2} - 1 ) d x}{x ^{3} 2 x ^{4} - 2 x ^{2} + 1} d x$ is

$2 2 - \frac{2}{x ^{2}} + \frac{1}{x ^{4}} + c$

$2 2 + \frac{2}{x ^{2}} + \frac{1}{x ^{4}} + c$

$\frac{1}{2} 2 - \frac{2}{x ^{2}} + \frac{1}{x ^{4}} + c$