Question

integrate this

∫cos⁷x dx

Answer 1

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$=>I= \displaystyle \int \ {cos}^{7} \: x \: dx$

$=>I = \displaystyle \int \: {cos}^{6} x \: cos \: x \: dx$

$=>I= \displaystyle \int(1 - {sin}^{2 \: } x) ^{3 \:} cos\: x \: dx$

$Let \: \sin x = t \\ and, cos \: dx = dt$

$=>I= \displaystyle \int (1 - t ^{2} ) ^{3} \: dt$

$=>I = \displaystyle \int(1 - {t}^{6} - {3t}^{2 \:} + {3t}^{4} ) \: dt$

$=>I = t - \frac{ {t}^{7} }{7} - {t}^{3} + \frac{3}{5} {t}^{5} + C$

$=>I= sin \: x - \frac{ {sin}^{7} }{7}x - sin ^{3} x + \frac{3}{5 \: } sin ^{5 } + C$