Differentiation and Integration (Trigonometric Functions) (1 Viewer)

BlueGas · Apr 28, 2015

I need help with question 5, how would I do this question?

InteGrand · Apr 28, 2015

BlueGas said:
I need help with question 5, how would I do this question?

$\frac{\mathrm{d} }{\mathrm{d} x}(\sin^3x)=3\sin^2 x \cos x$

$\Rightarrow \frac{1}{3}\frac{\mathrm{d} }{\mathrm{d} x}(\sin^3x)=\sin^2 x \cos x.$

$\therefore \int_0 ^{\frac{\pi}{2}}(\sin^2 x \cos x)\text{ d}x= \int_0 ^{\frac{\pi}{2}}\left( \frac{1}{3}\frac{\mathrm{d} }{\mathrm{d} x}(\sin^3x) \right) \text{ d}x$ (integrating both sides of the above)

$=\frac{1}{3} \int_0 ^{\frac{\pi}{2}} \frac{\mathrm{d} }{\mathrm{d} x}(\sin^3x) \text{ d}x$ (bring out the one-third)

$=\frac{1}{3} \left[\sin^3 x \right] _0 ^{\frac{\pi}{2}}$ (Fundamental Theorem of Calculus (note that the primitive of a derivative of a function is the function itself))

$=\frac{1}{3} (1-0)$

$=\frac{1}{3}.$

Differentiation and Integration (Trigonometric Functions) (1 Viewer)

BlueGas

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InteGrand

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