KCET2020MathematicsIndefinite Integration
The value of 1+x⁴ 1+x⁶ d x is
Options
- A⁻¹ x+ ⁻¹ x³+C
- B⁻¹ x+ 1 3 ⁻¹ x³+C
- C⁻¹ x- 1 3 ⁻¹ x³+C
- D⁻¹ x+ 1 3 ⁻¹ x²+C
Correct answer
B. ⁻¹ x+ 1 3 ⁻¹ x³+C
Step-by-step solution
We have, aligned 1+x⁴ 1+x⁶ d x =& 1+x⁴ 1+x⁶ 1+x² 1+x² d x =& 1+x²+x⁴+x⁶ (1+x⁶ ) (1+x² ) d x =& 1+x⁶ (1+x⁶ ) 1+x² d x+ x² (1+x² ) (1+x⁶ ) (1+x² ) d x =& d x 1+x² + x² 1+x⁶ d x =& ⁻¹ x+ x² 1+ (x³ )² d x =& ⁻¹ x+ d t / 3 1+t² aligned Put x³=t aligned 3 x² d x &=d t x² d x &= d t 3 &= ⁻¹ x+ 1 3 ⁻¹(t)+C &= ⁻¹ x+ 1 3 ⁻¹ (x³ )+C aligned