Question

If $f\left(x\right) = \int\limits^{sin\,x}_{2x}cos\left(t^{3}\right)dt$, then $f'{x}$ is equal to

• A. $cos(sin\, x)cos\,x-2cos(8x^3)$
• B. $sin(sin^3\, x)sin -2sin(8x^3)$
• C. $cos(cos^3\, x)cos\,x-2\,cos(x^3)$
• D. $cos(sin^3\, x)-cos(8x^3)$
• E. $sin(sin^3\, x)cos\,x-2\,sin(8x^3)$

Answer 1

Answer: (A)

Given, $f(x)=\int_{2 x}^{\sin x} \cos t^{3} d t$
Using Leibnitz's rule
$f^{\prime}(x)=\cos \left(\sin ^{3} x\right) \frac{d}{d x}(\sin x)$
$-\cos (2 x)^{3} \frac{d}{d x}(2 x)$
$=\cos \left(\sin ^{3} x\right)(\cos x)-\cos 8 x^{3}(2)$