∫ (d x/x(5)6(1+5 x(1)3) √1+4 x(1/3) equals (where ' c ' is a constant of integration)

Question

∫ (d x/x(5)6(1+5 x(1)3) √1+4 x(1/3) equals (where ' c ' is a constant of integration) (A) 6 tan -1(√(1+4 x(1/3)/x(1)3))+c (B) 3 ln |(1+√4+x-(1/3)/1-√4+x-(1)3)|+c (C) 6 cot -1(√(1+4 x(1/3)/x(1)3))+c (D) 3 ln |(1-√4+x-(1/3)/1+√4+x-(1)3)|+c

Tardigrade Admin · Accepted Answer

Correct answer is (c) 6 cot -1(√(1+4 x(1/3)/x(1)3))+c

Q. $\int \frac{d x}{x ^{\frac{5}{6}} ( 1 + 5 x ^{\frac{1}{3}} ) 1 + 4 x ^{\frac{1}{3}}}$ equals
(where ' $c$ ' is a constant of integration)

$6 tan^{- 1} (\frac{1 + 4 x ^{\frac{1}{3}}}{x ^{\frac{1}{3}}}) + c$

$3 ln ∣ ∣ \frac{1 + 4 + x ^{- \frac{1}{3}}}{1 - 4 + x ^{- \frac{1}{3}}} ∣ ∣ + c$

$6 cot^{- 1} (\frac{1 + 4 x ^{\frac{1}{3}}}{x ^{\frac{1}{3}}}) + c$

$3 ln ∣ ∣ \frac{1 - 4 + x ^{- \frac{1}{3}}}{1 + 4 + x ^{- \frac{1}{3}}} ∣ ∣ + c$

Correct answer is (c) $6 cot^{- 1} (\frac{1 + 4 x ^{\frac{1}{3}}}{x ^{\frac{1}{3}}}) + c$

Q. ∫x65​(1+5x31​)1+4x31​​dx​ equals (where ' c ' is a constant of integration)

6tan−1(x31​1+4x31​​​)+c

3ln∣∣​1−4+x−31​​1+4+x−31​​​∣∣​+c

6cot−1(x31​1+4x31​​​)+c

3ln∣∣​1+4+x−31​​1−4+x−31​​​∣∣​+c