Mathematics

Solve $$\displaystyle\int\limits_0^{\pi /2} {{{\sin }^4}x{{\cos }^3}xdx} $$ 


ANSWER

$$\dfrac {2}{35}$$


SOLUTION
$$\int_{0}^{\pi/2}\sin^{4} x\cos^{3} x dx$$
$$\int_{0}^{\pi/2}\sin^{4} x(1-\sin^{2} x)\cos x dx$$       $$\because \cos^{2} x=1-\sin^{2} x$$
$$\int_{0}^{\pi/2}(\sin^{4} x-\sin^{6} x)\cos x dx$$   
substitute $$\sin x=t$$  hence, $$dt=\cos x\ dx$$  
 and at $$x=0; \ t=\sin (0)=0 ; $$ at $$x=\pi/2; \ t=\sin(\pi/2)=1$$
$$\int_{0}^{1}(t^{4} -t^{6} ) dt$$ 
$$\bigg[\dfrac{t^{5}}{5}-\dfrac{t^{7}}{7}\bigg]_{0}^{1}$$ 
$$\bigg[\dfrac{1}{5}-\dfrac{1}{7} \bigg]=\dfrac{2}{35}$$
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Single Correct Medium Published on 17th 09, 2020
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