1.

Find ∫ (sin5 x - sin7 x)\(^{1\over2}\) dx1. \(\rm {7\over2}\sin^{7\over2} x\) + C2. \(\rm {7\over2}\cos^{7\over2} x\) + C3.  \(\rm {2\over7}\sin^{7\over2} x\) + C4.  \(\rm {2\over7}\cos^{7\over2} x\) + C

Answer» Correct Answer - Option 3 :  \(\rm {2\over7}\sin^{7\over2} x\) + C

Concept:

Integral property:

  • ∫ xn dx = \(\rm x^{n+1}\over n+1\)+ C ; n ≠ -1
  • \(\rm∫ {1\over x} dx = \ln x\) + C
  • ∫ ex dx = ex+ C
  • ∫ ax dx = (ax/ln a) + C ; a > 0,  a ≠ 1
  • ∫ sin x dx = - cos x + C
  • ∫ cos x dx = sin x + C

Substitution method: If the function cannot be integrated directly substitution method is used. To integration by substitution is used in the following steps:

  • A new variable is to be chosen, say “t”
  • The value of dt is to is to be determined.
  • Substitution is done and integral function is then integrated.
  • Finally, initial variable t, to be returned.

 

 

Calculation:

I = ∫(sin5 x - sin7 x)\(^{1\over2}\) dx

⇒ I = ∫[sin5 x (1 - sin2 x)]\(^{1\over2}\)dx

⇒ I = ∫ (sin5 x)\(^{1\over2}\)(cos2 x)\(^{1\over2}\)dx

⇒ I = ∫ sin\(^{5\over2}\) x cos x dx

Substituting sin x = t ⇒ cos x dx = dt

⇒ I = ∫ t\(^{5\over2}\) dt

⇒ I = \(\rm \left[t^{7\over2}\over{7\over2}\right]\) + C

⇒ I = \(\boldsymbol{\rm {2\over7}\sin^{7\over2} x}\) + C


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