If the value of integral displaystyle ∫ (d x/(x + √x2 - 1)2)=ax3-x+b(x2 - 1)(1/b)+C , (where, C is the constant of integration), then a× b is equal to

Question

If the value of integral displaystyle ∫ (d x/(x + √x2 - 1)2)=ax3-x+b(x2 - 1)(1/b)+C , (where, C is the constant of integration), then a× b is equal to (A) 1 (B) (4/9) (C) 2 (D) (9/4)

Tardigrade Admin · Accepted Answer

displaystyle ∫ (x - √x2 - 1)2 d x= displaystyle ∫ x2 + (x2 - 1) - 2 x √x2 - 1 d x =(2 x3/3)-x+(2/3)(x2 - 1)(3/2)+C ⇒ a=b=(2/3)

Q. If the value of integral $\int \frac{d x}{( x + x ^{2} - 1 ) ^{2}} = a x^{3} - x + b (x^{2} - 1)^{\frac{1}{b}} + C$ , (where, $C$ is the constant of integration), then $a \times b$ is equal to

$1$

$\frac{4}{9}$

$2$

$\frac{9}{4}$

$\int (x - x^{2} - 1)^{2} d x = \int x^{2} + (x^{2} - 1) - 2 x x^{2} - 1 d x$
$= \frac{2 x ^{3}}{3} - x + \frac{2}{3} (x^{2} - 1)^{\frac{3}{2}} + C$ $\Rightarrow a = b = \frac{2}{3}$

Q. If the value of integral ∫(x+x2−1​)2dx​=ax3−x+b(x2−1)b1​+C , (where, C is the constant of integration), then a×b is equal to

1

94​

2

49​

∫(x−x2−1​)2dx=∫x2+(x2−1)−2xx2−1​dx =32x3​−x+32​(x2−1)23​+C ⇒a=b=32​

Q. If the value of integral $\int \frac{d x}{( x + x ^{2} - 1 ) ^{2}} = a x^{3} - x + b (x^{2} - 1)^{\frac{1}{b}} + C$ , (where, $C$ is the constant of integration), then $a \times b$ is equal to

$1$

$\frac{4}{9}$

$2$

$\frac{9}{4}$

$\int (x - x^{2} - 1)^{2} d x = \int x^{2} + (x^{2} - 1) - 2 x x^{2} - 1 d x$
$= \frac{2 x ^{3}}{3} - x + \frac{2}{3} (x^{2} - 1)^{\frac{3}{2}} + C$ $\Rightarrow a = b = \frac{2}{3}$