Question

can anybody do this question of integration

Answer 1

Let I=∫cos9xsinxdxI=∫cos9⁡xsin⁡xdx

=∫cos8xcosxsinxdx=∫cos8⁡xcos⁡xsin⁡xdx

=∫(cos2x)4cosxsinxdx=∫(cos2⁡x)4cos⁡xsin⁡xdx

=∫(1−sin2x)4cosxsinxdx=∫(1−sin2⁡x)4cos⁡xsin⁡xdx

Let sinx=usin⁡x=u

⟹du=cosxdx⟹du=cos⁡xdx

⟹I=∫(1−u2)4udu⟹I=∫(1−u2)4udu

=∫1−4u2+6u4−4u6+u8udu=∫1−4u2+6u4−4u6+u8udu

=∫1u−4u+6u3−4u5+u7du=∫1u−4u+6u3−4u5+u7du

=ln|u|−2u2+3u42−2u63+=ln⁡|u|−2u2+3u42−2u63+u88+Cu88+C

=ln|sinx|−2sin2x+3sin4x2−2sin6x3+sin8x8+C=ln⁡|sin⁡x|−2sin2⁡x+3sin4⁡x2−2sin6⁡x3+sin8⁡x8+C
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