Question

The value of $\underset{ h \rightarrow 0}{\text{Lim}} \frac{\int\limits_{ a }^{ x + h } \frac{(\sin t )^{2014}}{ t ^{2013}+1} dt -\int\limits_{ a }^{ x } \frac{(\sin t )^{2014}}{ t ^{2013}+1} dt }{ h }$ is equal to

• A. $\frac{2013(\sin x)^{2013}}{2014(x)^{2013}+1}$
• B. $\frac{(\sin x)^{2015}}{(x)^{2014}+1}$
• C. $\frac{(\sin x)^{2014}}{x^{2013}+1}$
• D. $\frac{2014(\sin x)^{2013}}{2013\left(x^{2012}+1\right)}$

Answer 1

Answer: (C)

$\underset{ h \rightarrow 0}{\text{Lim}} \frac{\int\limits_a^{x+h} \frac{(\sin t)^{2014}}{t^{2013}+1} d t-\int\limits_a^x \frac{(\sin t)^{2014}}{t^{2013}+1} d t}{h}$
differentiate w.r.t. h
$\underset{ h \rightarrow 0}{\text{Lim}} \frac{\frac{(\sin (x+h))^{2014}}{(x+h)^{2013}+1}-0}{1}=\frac{(\sin x)^{2014}}{x^{2013}+1}$