Question

Integral of (sin x)^4 ×(cos x)^3 dx​

Answer 1

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:Let \: A\: = \: \displaystyle\int\sf {sin}^{4}x \: {cos}^{3}x \: dx$

$\rm \: \: A = \: \: \:\displaystyle\int\sf {sin}^{4}x \: \times {cos}^{2}x \times cosx \: dx$

$\rm \: \: A = \: \: \:\displaystyle\int\sf {sin}^{4}x \: \times(1 - {sin}^{2}x) \times cosx \: dx$

$\red{\bigg \{ \because \: {sin}^{2}x + {cos}^{2}x = 1 \bigg \}}$

Now, we use method of Substitution,

$\red{\rm :\longmapsto\:Let \: sinx = y} \\ \red{\rm :\longmapsto\:\dfrac{d}{dx} sinx = \dfrac{d}{dx}y} \\ \red{\rm :\longmapsto\:cosx = \dfrac{dy}{dx} \: \: \: \: \: \: \: } \\ \red{\rm :\longmapsto\:cosx \: dx \: = dy \: \: \: }$

On substituting these values we get

$\rm \: A= \: \: \displaystyle\int\sf {y}^{4} (1 - {y}^{2}) \: dy$

$\rm \: A= \: \: \displaystyle\int\sf ( {y}^{4} - {y}^{6}) \: dy$

$\rm \: A= \: \: \dfrac{ {y}^{4 + 1} }{4 + 1} - \dfrac{ {y}^{6 + 1} }{6 + 1} + c$

$\rm \: A= \: \: \dfrac{ {y}^{5} }{5} - \dfrac{ {y}^{7} }{7} + c$

$\rm \: A= \: \: \dfrac{ {(sinx)}^{5} }{5} - \dfrac{ {(sinx)}^{7} }{7} + c$

$\rm \: A= \: \: \dfrac{ {sin}^{5}x }{5} - \dfrac{ {sin}^{7} x}{7} + c$

Additional Information :-

$\green{\boxed{ \bf \:\displaystyle\int\sf kdx=kx + c }}$

$\green{\boxed{ \bf \:\displaystyle\int\sf sinx \: dx= - \: cosx + c }}$

$\green{\boxed{ \bf \:\displaystyle\int\sf cosx \: dx= \: sinx + c }}$

$\green{\boxed{ \bf \:\displaystyle\int\sf secx \: tanx\: dx= \: secx + c }}$

$\green{\boxed{ \bf \:\displaystyle\int\sf cosecx \: cotx\: dx= - \: cosecx + c }}$

$\green{\boxed{ \bf \: \displaystyle\int\sf {x}^{n} \: dx = \dfrac{ {x}^{n + 1} }{ n+ 1} + c}}$

$\green{\boxed{ \bf \: \displaystyle\int\sf {e}^{x} \: dx = {e}^{x} + c}}$

$\green{\boxed{ \bf \: \displaystyle\int\sf tanx \: dx = log(secx) + c}}$

$\green{\boxed{ \bf \: \displaystyle\int\sf cotx \: dx = log(sinx) + c}}$

$\green{\boxed{ \bf \: \displaystyle\int\sf secx \: dx = log(secx \: + \: tanx) + c}}$

$\green{\boxed{ \bf \: \displaystyle\int\sf cosecx \: dx = log(cosecx \: - \: cotx) + c}}$