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Manipal MET2022MathematicsIndefinite Integration

If f(x) d x=g(x) , then x^9 f (x^5 ) d x is equal to

Options

  1. A1 5 (x^5 g (x^9 )-4 g(x) d x+C .
  2. B1 5 [x^9 g (x^5 )- 1 5 x^4 g (x^5 ) d x+C .
  3. C1 5 [g (x^9 )+ g (x^5 ) d x ]
  4. Dx^5 5 g (x^5 )- x^4 g (x^5 ) d x+C

Correct answer

D. x^5 5 g (x^5 )- x^4 g (x^5 ) d x+C

Step-by-step solution

Given that, f(x) d x=g(x) Let I= x^9 (x^5 ) d x Put x^5=t 5 x^4 d x=d t aligned & = 1 5 t f(t) d t & = 1 5 [t f(t) d t- [f(t) d t] . aligned [integrating by part] aligned & = 1 5 [t g(t)- g(t) d t ] & = 1 5 [x^2 g (x^5 )-5 g (x^5 ) x^4 d x ]+C & = x^5 5 g (x^5 )- x^4 g (x^5 ) d x+C aligned

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