First slide

Methods of integration

Question

$\int \sin^{\frac{5}{2}} x . \cos^{3} x d x =$

Easy

A

$2 \sin^{\frac{7}{2}} x (\frac{1}{7} + \frac{1}{11} \sin^{2} x) + c$
B

$2 \sin^{\frac{7}{2}} x (\frac{1}{7} - \frac{1}{11} \sin^{2} x) + c$
C

$3 \sin^{\frac{7}{2}} x (\frac{1}{5} - \frac{1}{9} \sin^{2} x) + c$
D

$3 \sin^{\frac{7}{2}} x (\frac{1}{7} - \frac{1}{11} \sin^{2} x) + c$

Solution

$\int \sin^{\frac{5}{2}} x [1 - \sin^{2} x] \cos x d x$
      $= \int t^{\frac{5}{2}} [1 - t^{2}] d t$

      $= \int (t^{\frac{5}{2}} - t^{\frac{9}{2}}) d t$

      $= \frac{2}{7} [t^{\frac{7}{2}}] - \frac{2}{11} t^{\frac{11}{2}} + C$

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