Value of the definite integral ∫ limits-1 / 21 / 2( sin -1(3 x-4 x3)- cos -1(4 x3-3 x)) d x

Question

Value of the definite integral ∫ limits-1 / 21 / 2( sin -1(3 x-4 x3)- cos -1(4 x3-3 x)) d x (A) 0 (B) -(π/2) (C) (7 π/2) (D) (π/2)

Tardigrade Admin · Accepted Answer

Note that in (-(1/2), (1/2)), sin -1(3 x-4 x3)=3 sin -1 x and cos -1(4 x3-3 x)=2 π-3 cos -1 x hence f(x)=3 sin -1 x-2 π+3 cos -1 x=-(π/2) ∴ I=-(π/2) ∫ limits-1 / 21 / 2 d x=-(π/2)

Q. Value of the definite integral $- 1/2 \int 1/2 (sin^{- 1} (3 x - 4 x^{3}) - cos^{- 1} (4 x^{3} - 3 x)) d x$

0

$- \frac{π}{2}$

$\frac{7 π}{2}$

$\frac{π}{2}$

Note that in $(- \frac{1}{2}, \frac{1}{2}), sin^{- 1} (3 x - 4 x^{3}) = 3 sin^{- 1} x$ and $cos^{- 1} (4 x^{3} - 3 x) = 2 π - 3 cos^{- 1} x$ hence
$f (x) = 3 sin^{- 1} x - 2 π + 3 cos^{- 1} x = - \frac{π}{2}$
$∴ I = - \frac{π}{2} - 1/2 \int 1/2 d x = - \frac{π}{2}$

Q. Value of the definite integral −1/2∫1/2​(sin−1(3x−4x3)−cos−1(4x3−3x))dx

0

−2π​

27π​

2π​

Note that in (−21​,21​),sin−1(3x−4x3)=3sin−1x and cos−1(4x3−3x)=2π−3cos−1x hence f(x)=3sin−1x−2π+3cos−1x=−2π​ ∴I=−2π​−1/2∫1/2​dx=−2π​

Q. Value of the definite integral $- 1/2 \int 1/2 (sin^{- 1} (3 x - 4 x^{3}) - cos^{- 1} (4 x^{3} - 3 x)) d x$

$- \frac{π}{2}$

$\frac{7 π}{2}$

$\frac{π}{2}$

Note that in $(- \frac{1}{2}, \frac{1}{2}), sin^{- 1} (3 x - 4 x^{3}) = 3 sin^{- 1} x$ and $cos^{- 1} (4 x^{3} - 3 x) = 2 π - 3 cos^{- 1} x$ hence
$f (x) = 3 sin^{- 1} x - 2 π + 3 cos^{- 1} x = - \frac{π}{2}$
$∴ I = - \frac{π}{2} - 1/2 \int 1/2 d x = - \frac{π}{2}$