Let for all real

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

Let

f(t)=\sqrt{t^{2}+1}

for all real

t

. Define the function

G(x)=\int_{0}^{\sin x} f(t)\,dt

for

x

in the interval

\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]

. (a) Calculate

G'(x)

. (b) Evaluate

G'\!\left(\dfrac{\pi}{6}\right)

. (c) Determine the equation of the tangent line to the graph of

y=G(x)

x=\dfrac{\pi}{6}

. Express your answer in point–slope form.