Question

Find ∫ (sin⁵ x - sin⁷ x)^{\(^{1\over2}\)} dx
1. \(\rm {7\over2}\sin^{7\over2} x\) + C
2. \(\rm {7\over2}\cos^{7\over2} x\) + C
3.  \(\rm {2\over7}\sin^{7\over2} x\) + C
4.  \(\rm {2\over7}\cos^{7\over2} x\) + C
5. None of these

Answer 1

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