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Integrals

Question
CBSEENMA12036062

If ∫f x dx =Ψ (x),then ∫x5 f(x3)dx is equal to

  • 1 third space open square brackets straight x cubed straight psi left parenthesis straight x cubed right parenthesis minus integral straight x squared space straight psi left parenthesis straight x cubed right parenthesis dx close square brackets space plus straight C
  • 1 third space straight x cubed straight psi left parenthesis straight x cubed right parenthesis minus 3 integral straight x squared space straight psi left parenthesis straight x cubed right parenthesis dx space plus straight C
  • 1 third straight x cubed straight psi left parenthesis straight x cubed right parenthesis minus integral straight x squared space straight psi left parenthesis straight x cubed right parenthesis dx space plus space straight C
  • 1 third open square brackets straight x cubed straight psi left parenthesis straight x cubed right parenthesis minus integral straight x squared space straight psi left parenthesis straight x cubed right parenthesis dx close square brackets plus space straight C

Solution

C.

1 third straight x cubed straight psi left parenthesis straight x cubed right parenthesis minus integral straight x squared space straight psi left parenthesis straight x cubed right parenthesis dx space plus space straight C

Given, ∫f(x) dx = Ψ (x)
Let l = ∫x5f(x3)dx
Put x3 = t 
⇒ x2dx = dt/3 ...... (i) 
From space equ space left parenthesis straight i right parenthesis
therefore space l space equals space 1 third integral straight t space straight f left parenthesis straight t right parenthesis space dt space equals space 1 third left square bracket straight x cubed space straight psi left parenthesis straight x cubed right parenthesis minus 3 integral straight x squared straight psi left parenthesis straight x cubed right parenthesis dx right square bracket space plus straight C space
equals space 1 third space straight x cubed space straight psi left parenthesis straight x cubed right parenthesis space minus integral straight x squared space straight psi left parenthesis straight x cubed right parenthesis space dx space plus space straight C space

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